General Relativity | Lecture 6

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General Relativity | Lecture 6
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(October 29, 2012) Leonard Susskind presents the physics of black holes including the event horizon, the photon sphere, and the singularity. This series is the fourth installment of a six-quarter series that explore the foundations of modern physics. In this quarter, Professor Susskind focuses on Einstein's General Theory of Relativity and in this lecture Leonard Susskind moves the course into discussions of gravity and basic gravitational fields.
Funkgerät Cosmic distance ladder Gravitational singularity Stellar atmosphere Rubber stamp Force Particle Watch Roll forming Temperaturabhängiger Widerstand Universe Cartridge (firearms) Mass Book cover Material
Trajectory Mechanic Short circuit Mass Video TARGET2 Force Mobile phone Radial engine Noise figure FACTS (newspaper) Particle
Girl (band) Quality (business) Geokorona Scale (map) Hochbahn Thermodynamic equilibrium Light Particle Group delay and phase delay Fighter aircraft Roll forming Aerial tramway Mechanic Year Railroad car FACTS (newspaper) Specific weight Direct current Autumn Current density Cartridge (firearms) Newtonsche Flüssigkeit Mass Plane (tool) Noise figure Flatcar Orbit Ruler Funkgerät Cosmic distance ladder Gaussian beam Trajectory Clock Measurement Interval (mathematics) Northrop Grumman B-2 Spirit Star Weather front Cell (biology) Band gap Arbeitszylinder Be star Yardstick Video Negation Brewster's angle Angeregter Zustand Kontraktion Typesetting Surfing Redshift Transmission tower Winter Electronic component Classical mechanics Finger protocol Black hole Angle of attack Nyquist stability criterion Sizing Standard cell Stardust (novel) Remotely operated underwater vehicle Miner Klappmesser Ground (electricity)
Girl (band) Energiesparmodus Acceleration Trajectory Sound Speed of light Geokorona Amplitude Force Radial engine Angular velocity Particle Mitsubishi A6M Zero Bird vocalization Roots-type supercharger Spare part Mail (armour) Videotape Electronic component Year Drehmasse FACTS (newspaper) Kopfstütze Heat Angle of attack Direct current Fahrgeschwindigkeit Cartridge (firearms) Mass Satellite Gas turbine Noise figure Miner Orbit Radioactive decay
Girl (band) Oncotic pressure Energiesparmodus Photocopier Mechanical fan Yellow Effects unit Photography Destroyer Geokorona Radial engine Thermodynamic equilibrium Rain Particle Limiter Artillery battery Tool bit Combined cycle Photon Year Minute Hydrogen atom Source (album) Missile Fahrgeschwindigkeit Particle physics Cartridge (firearms) Mass Food storage Noise figure Orbit Ruler Alternator Funkgerät Hot working Cosmic distance ladder Speed of light Measurement Angular velocity Roots-type supercharger Band gap Halo (optical phenomenon) Radial velocity Spare part Nanotechnology Homogeneous isotropic turbulence Stagecoach Measuring cup Drehmasse Cogeneration Finger protocol Gentleman Nyquist stability criterion Miner Stock (firearms)
Capital ship Analog signal Acceleration Photography Active laser medium Thermodynamic equilibrium Light Particle Ring (jewellery) Bird vocalization Pattern (sewing) Cylinder block Ship breaking Audio frequency DVD player Photon Year Casting defect FACTS (newspaper) Cosmic microwave background radiation Focus (optics) Direct current Rolling (metalworking) Fahrgeschwindigkeit Cartridge (firearms) Mass Noise figure Orbit Funkgerät Cosmic distance ladder Hot working Gravitational singularity Literary fragment Atomhülle Northrop Grumman B-2 Spirit Star Mask Spiral galaxy Elektronenbeugung Radial velocity Spare part Bicycle lighting Spaceport Negation Rail transport operations Leistungsanpassung Material Catadioptric system Measuring cup White Drehmasse Classical mechanics Black hole Angle of attack Gentleman Paint Standard cell Firearm
Trajectory Analog signal Cosmic distance ladder Acceleration Kickstand Netztransformator Differential (mechanical device) Measurement Magic (cryptography) Foot (unit) Towing Watch Gasbohrloch Screwdriver Bird vocalization Turning Sunrise Orbital period Veränderlicher Stern Month Runic calendar Negation Tape recorder Watercraft rowing Jukos Reference work Programmable Array Logic Cut (gems) Tool Year Car seat Railroad car Microscope Drehmasse Kopfstütze Focus (optics) Spant Black hole Angle of attack Direct current Cartridge (firearms) Deck (ship) Noise figure Water vapor Flatcar Ground (electricity)
Hot working Acceleration Trajectory Clock Speed of light Gravitational singularity Day Measurement Particle Cell (biology) Watch Bird vocalization Apparent magnitude Filing (metalworking) Halo (optical phenomenon) Wednesday Orbital period Video Watercraft rowing Atmosphere of Earth Quark Weather Typesetting Bauxitbergbau Vehicle armour Tool Steel Key (engineering) Crystal habit Car seat Bending (metalworking) Year Ground station Drehmasse FACTS (newspaper) Yacht Finger protocol Black hole Gentleman Direct current Autumn Hour Noise figure
Hot working Acceleration Clock Speed of light Gravitational singularity Bomb Day Measurement Crystal structure Foot (unit) Bird vocalization Roll forming Alcohol proof Musical ensemble Will-o'-the-wisp Burst (band) Photon Automobile platform Black Lead Yacht Black hole Angle of attack Standard cell Series and parallel circuits Marble sculpture Stock (firearms) Orbit Near field communication Firearm
Gang Weight Spaceflight Radial engine Light Rain Group delay and phase delay Flashlight Wavelength Continuous track Sunrise Limiter Rail profile Audio frequency Sensor Photon Year Source (album) FACTS (newspaper) Direct current Autumn Signal (electrical engineering) Fahrgeschwindigkeit Cartridge (firearms) Mint-made errors Mass Electronic media Book design Noise figure Orbit Funkgerät Spread spectrum Trajectory Cosmic distance ladder Hot working Clock Speed of light Gravitational singularity Day Atomhülle Automated teller machine Kosmischer Staub Star Brickyard Orbital period Material Angeregter Zustand Surfing Array data structure Cylinder head Lead Hydraulic accumulator Drehmasse Hot isostatic pressing Kopfstütze Microwave Black hole Game Sizing Gentleman Standard cell Wind wave Manned mission to Mars Firearm
Stecher University OK
let's let's begin with they were in the midst of discussing the properties of the Watch Hill Blackall with barely began can we really barely began sorrow at the review quickly the Swat chilled solutions 1st of was idealized solution that idealized in the sense that on the new Tony and F. rarest vs. brokers as equals and end G all our squared that's an idealization Thyssen an idealization under the assumption that the mass creating the creating the gravitational field is a point at the center of a point amassed creates a gravitational field like that if the masses spread out that of course we don't really believe this to be the case in the interior of where the and the interior of the Earth is not correct but this show In X exterior this throughout beyond the surface of Europe at least if we ignore the atmosphere and the other forms of gravitating mass Isoetes an idealization . it looked on only mechanic should never really ever reach a point mass real materials have asserted stiffness who cleared the squeeze on them but to condemn the squeeze Tula infinitely small radios they just have a certain Her resistance to being squeezed gravity is always so strong that is always strong even the Vold the gravitational field appears to get very very large short distances the nature of materials old such that if you squeeze stamp arbitrarily small distances it would bring back not sold general theory of relativity with example tonight cover way in which such that matter gets too close to the singularity it gets sucked in In a way that doesn't happen in the Tony graf Newtonian gravity Pa for example in the tiny gravity for a Newtonian physics in general here if you have a force center like and you put the lives idealized an idealized forced their privacy and you put in a particle which is moving straight for the force center and with infinite
precision yes indeed it was at the center but even if you have the littlest the littlest deviation away from an ideal perfect perfection and the trajectory of the particle it will not hit the center animal swirling around it effectively there's a kind of forest repelling it from the center law that forces called sent cervical force from mobile force if you're a team is not 1st the will have a little bit of angular momentum and because of that angular momentum they will effectively be said tropical forests and cervical force will keep it from going through the center of in the county in mechanics even lower Newtonian gravity even Though the center pulls very hard when you get close to it nevertheless it is infinitely unlikely in falling point particle actually hit the center not sold in 9 of the general theory of relativity general theory of relativity gravity is even stronger itself stronger than overwhelms video the barrier and imposed anything into a center I think the fact that also but set in they should know well there's if this is true this is true love but didn't know the general theory of relativity you deleted young into pocket lets a right which watch over the short film metric we did not authorize the world and allies than later come back out tell you what equations misfortune metric is a solution disillusioned Einstein's field equations but figures a target this is something later will want to deride the did say specific metric specific geometry whose properties of the analyzes that don't described by a proper time to proper time interval every family for its 1 2 em a mass of the object the Viet times of the gravitational constant provided by the radial
coordinate this is just twice the potential the
potential energy about unit messed particle moving and distance are best friends deeply squared minus 1 or the same things 1 minus and G or what are the odds squared and notice that when you pass Our equals to energy as you can't even center this change signed but so is there so that will always be warned coordinated with a positive sign here and 3 other coordinates with a negative sign what's right and the other 3 coordinates the other 3 coordinates would-be are squared and expressed this as deal maker squared the Omega square Frederick here the Omega square it is just the metric units spirits just a metric of sphere in now Pola quarter renowned as news and as and polar angle so it's just traders squared plus co-starring of fate squared defy squared was escorted back armed sometimes we may only be interested in 1 of these angles for example would be interested in the problem of the orbiting particle if the auditing is around the equator back there forget 5 altogether no we can forget sorry we can forget a we only 1 angle we only need 1 angle only to angles and quality fire square incidentally cosigned favor at the equator is just 1 and so happy equated this just be defy Square OK sets that's his mentor Kieran let go picture of a Westrup picture 1st would draw the Timex climaxes I would do a vertically there was a time axis horizontal space would only have Till directions space American draw so that figure flat planes intersecting plan equals 0 of course is really a three-dimensional planes but the fact that the they had all our peoples to energy something is happening this of course center here could thought Oracle 0 services are equal 0 oracle 0 % that we would do exactly the same thing if we were talking about the new Tony and point particle religious place a particle center then somewheres are at or equals to win the something happening or not happening some funniness in many cases mathematical structure of its struck a cylinder surrounding Oracle 0 it's called ladies and it's called the horizon of the black hole we understand why as we go along the horizon of the black hole or to switch out beyond that very far away let's look at what's going on very far away very far away 1 over 2 ng that's negligible if we got a very large distances 102 and more of our becomes very small so this just becomes squared this just becomes miners DR squared minus our squared the Omega square that a flat at the flat space full-court flat space time so far away it's just flat space time but as we move in for Oracle's toe and GE something happens and the geometry is characteristically different now it is true that something will look bad and Oracle's to Ng 1st of all a coefficient of becomes era but even worse the coefficient of Dr squared becomes infinite so something's blowing up to see that nothing of any real physical significance is happening at that point on the other hand and Oracle 0 0 article 0 the denominator he had become 0 1 minus 20 of our becomes infinite and you might ask is something real going on that article's toe injury not sorry article 0 India something real was going on 0 and what's going on Oracle 0 is the curvature is becoming that's the thing that's going on nothing special no large curvature is happening no enormous deviation from flat space is happening at at articles to Angie by Oracle 0 all hell was breaking loose the Pruitt of calculated the curvature can't find out just how curve this geometry is we find all of its components become infinite write articles 0 with another later described that curvature means tidal forces anything falling where would hit Oracle 0 would experience infinitely strong tidal forces and b . 2 pieces who would be stretched radially squeeze the angular direction an infinite amount so are equal 0 is a real place in other words a real love serious disaster Oracle's to Angie nothing special OK we talk a little bit about radios in falling things of this remind you of what the conclusion was I don't need to derive this again because we never see exactly how it works in terms of pictures when that didn't need the equations to see how it works it takes an infinite time for something falling in the past articles to that means in the who have drawn he s kind surfaces of constant kind horizontal the kind axis vertical if something more falling in it would look like server fall in Indian star faraway fault the black black all of the horizon murders and it it would accelerate for a while I want try drug accelerating Irma for that but as a closer and closer to the horizon Edward askSam told getting closer and closer but never quite getting there if this object had 80 will include there was a stick falling in the a long way if meterstick falling in a long way the front the meterstick might look like that the back end of the meter stick would do exactly the same thing over fall and it too would never across the horizon and so the only way to make sense out of this sale leased in these coordinates the gap In all are between the front then the need stick in the back end of stick shrinks 0 this is actually a form of Lawrence contraction has something falls in even tho it appears to be slowing down the momentum is increasing and Lawrence contracting but also think about 8 o'clock let's imagine that this meterstick was not just a meter state those o'clock how to build a clerk of the stay well you could put to mirrors through either in the video stick and allow all like to go back and forth and back and forth in you simply count the number of our pulses of light beams that go back and forth that would be 1 way of making a radio a clerk at avarice but clerks as they move however they're moving they don't kick off this kind of time we kick off proper time that's meaning of the proper time the proper time is thinking of moving Clark and fix the moving Clark makes from some starting point this time over here that's called Coordinate Time is just the coordinate that we using to describe Riyadh our the entire system notice that if borrowers very far away supposing will weigh have hiya not moving at all not be enough moving out with a deep Tal it is equal to do it at least high approximation because 1 2 is 1 of them is far away proper time and ordinary current are saying very far away factors just basically 1 indeed Tal
TO of the same thing and we're not moving so the R and 0 then Clark moving very far away will really just measures DT but as we get closer and closer in for a given amount of D D gets smaller and smaller surfer example here's a given amount of d of of time transpiring time I mean little to a given amount of wouldn't be time coordinate current trends buyers between this surface and that surface over here from much proper time as you get closer and closer the proper time the trams fires along this trajectory he gets smaller and smaller cell while it appears that the amount of TV that takes a fully through the black infinite we could ask how much how evolves by the time and in falling object eventually arrives at the horizon and that is finite began with going to see this pictures received this without doing any Mathematica abruptly India calculation or see it just from simple diagrams but leaders see why this might be or even tho the time or time interval he is getting longer longer because town is shrunken by comparison with loss his attempts in any case that the amount of time along 1 of these trajectories here is finite until you cross the boundary now somebody falling in with a clock do they expert with or without a clock ordinary human beings do they experience of my experience I need experience in the usual sense the experience an infinite time before they get to the horizon of finite and the answer is they experienced the proper your hard as a clerk your brain as a o'clock every kind keep you'll have will experience the same shift or a relative size scale for the beating of a clock tower and we the beating of Clark's very very far away so it'd be interesting to know whether or not the proper time that it takes the fall life Aaronson how we could solemnly equations could resolve equations we can calculate the winter girls but fortunately we don't really have to the pictures will take care of it before I get post pictures Norris you have solved the problem that does not have to the floor I have to do with somebody falling into the black hole but has to do with another question or emitting the black all orbiting the black also means well it's a little hard to draw orbiting the black all means corkscrew going around this time gross forward but let's take the case of a circular orbit a circular orbit means that if we projected the orbit onto a onto just are the space without worrying about time that they orbit would just be a circular Robert so grew looking down at this thing we're looking from the future down at the center Group see Oracle 0 we would see some Star Circle corresponding just watch over s articles to Oracle Sean due over here and we would say orbiting a particle going around take a case of a circular orbit circular orbits are of course particularly simple we figure out what a circular orbits yes we can but we're show you just 1 specific kind of circular orbit this but the particular kind of circular orbit we talk about Izzy or a bit of a light as you might expect and as you probably know that a light ray use again looked strong grow another picture of this horizon small I haven't seen the size of the horizon just looking at it from a larger distance alike Brady goes by we know from Einstein from a number of different places but the library will when it goes by our B of the massive object I did show you want warning that the equivalence principle of accelerating elevator does tell you that a light green moving across the elevator Mozilla Raiders accelerating will bend bandaged trajectory that sustained he gravity Ben's trajectories end fact obviously as you moving closer and closer that relies Ray will get more and more White because gravity becomes stronger as you move in that's what shows radius is really whips around and at some point it's not as further out than the horizon there exists a there exist trajectories which just go right around circular trajectories which just go right around the gravitating object that this can only happen this will not happen near the surface of the year you have the school easier down to practically become a black hole of to squeeze a way way down past itch what you'll radius incidentally an object of the mass of the earth would have assured children use of about 70 sort of awards the earth would not do anything like this on was was squeezed down of diameter of the young of the peanut so as not to worry in Athens you write raised is a circle around the Earth but nevertheless the very interesting to try and see if we can understand the orbiting libraries the calculation intensive here because I wanted to show you set of things that you do the calculate the circular orbit where do it by using the rules of classical mechanics things a we've already set up in previous quarters and harm the reason I'm doing it this way but the reason I'm doing it this is because the only easy way to do it and even it is not so easy but that it is worth the point that the things we learned before about classical mechanics for example are not disconnected from what we're doing now we're going to use classical mechanics and some consternation laws much more than conservation laws just a little bit more are the basic principle of stability that freezes that basic principle of equilibrium that equilibrium happens at extreme low points are at their stationary points of potential energy use that will be used mechanics object or buildings but not object the Newtonian object of object moving with this this magical here OK so what we talked about this last time her back over it but a lot of great length and injury the calculation we start with the action whenever you working out equations of motion for anything is almost always easiest to start with the action principle the action principle From the action principle arrival of Grosjean and from the Lagrangian arrived the a Lagrange equations of motion all or some other trek from going to do quite that the was conservation laws alright so we get we start with we start with this metric here and the role for the action it is equally the actions of a moving car of action of moving particle is a parameter from event the parameters of course a massive of particle is is necessary to give action was a you want the units fractionally abutting 40 units of a grand you Una energy use of action our energy terms time fighters of the others are going to give the right units has gotta be a mess out front the study energy times time messes
energy are not going to fool around with the speed of light will leave the speed of light equipped 1 for tonight U.S. elections are met times time energy times time so his energy and what rest of it the rest of it is the proper time of the orbit between 1 . 7 other point another words the Inter will T. D Tao along the trajectory from 1 from an initial . 0 Final but they did not know is the square root of what appears here so but let's rewrite it cow square root of family gives us cannot keep writing this thing over and over in this thing over and over and when give them some new names recall the subject asks court script death and 1 all over at all costs could G the reason I'm using script is because In order for not right this thing equal this them they want to use just ordinary Jedi so I called the scripted but then I said well at close was the death OK so this a script deference script its grip death times squared minus script GE times DR squared and then finally miners are squared now afford Omega square we're talking about a particle moving on a trajectory let's assume that these trajectory is a circle parameterized my a single angle it could be the angle of around the equator was all just call it miners are squared status square babies defy square the matter you it doesn't matter reverts polar wrangle over or the azimuthal angle of court dictator squared that's from a trajectory moving around with scored the equator of the year Of the sphere and some other words this angle here say Our Square favor square now what we do it's always the same thing divided by D C squared inside the square root and then multiplied by DT outside the square same Soviet-era the DT square at this are squares good become br by squared so that's just calls are not squared the radial component of a velocity squared this becomes by DG Square which is just say that about squared the angular velocity is the angular velocity but now I have to put back DT I removed decay squared but from under the square root this make that up by multiplying by GTE here f ng functions of all are the moment I want specified and let's just right there and don't remember the G is 1 already as it happens G 1 at Tour for a circular orbit are divers equal to 0 for a circular orbit the wounded simpler because for a circular orbit are about equal to 0 but we don't want quite go they yet I so early off the Lagrangian Lagrangian is just the thing inside the other thing inside integral Granzien scored script out minors and contains the fact miners and Thailand a square root path miners GE looked that as the GE are about squared minus are square a square what I want from there so I want to see direct conservation laws for this problem the conservation laws ah Energy Energy is always a conservation law any other conservation didn't think of orbiting satellites Hang momentum angular momentum angular momentum is just the moment the conjugate fate remember the definition of a momentum and Lagrangian mechanics if we have a coordinate called Q any court called Q. the momentum associated with Q just equal to derivative of Lagrangian with respect to Q . receiving them over and over again here it is again I angular momentum and unfortunately angular momentum is also called sales folklore not script out the angular momentum is just equal to the derivative logo but the red tape below grungy and with respect the . they were just applying this equation when she was equal to figure and that is equal to 0 although the square root like other continuously right the square root over and over again just call square root without running anything in here write square root you keep in mind that the square root of what's in here at times the derivative of what's inside with they've thought and that just give us and ah school where they got that's the angular momentum and the importance of the angular momentum is a constant the importance of the angular momentum is a country that's also proportional to the Mets I'm going to write it as a mail kind is something I'll call Greek Orlando when a factor at the Mass and what's left over call land was when that land is everything in here with the exception of the Mets so we could write this part of this divided by a L of our provided by and remove from heat and removed him from here but so here we have an expression for a quantity which is conserved its concerned because the angular momentum is conserved and it's a function of faded an aura about for a circular ordered it simplifies a little bit because are about equal to 0 but let's not do the circular orbit yet but calculate this is this it is well market Italy is the momentum associated with angle another momentum and the other momentum is associated with are the other quarter to quarter Nixon's problems are a trader and the find tiara we have to differentiate the Lagrangian with respect that's also equal to sum of zeros at-bats and a script GE are I think provided by the same square root same square-rigger think that's right br so we have are Babbitt or not is not considered does not considered their forces radial forces on their radial forces there accelerations along the radial axis of accelerations along we radioactive PR is not concerned but what is considered the energy somewhere I don't yet the general expression for the energy the general expression for the energy is girl Hamiltonian the classical Hamiltonian remind you what it's given by the somewhat over the various coordinates are the momentum kind of velocity associated with that coordinated the words time derivative some of the coordinates in this case there only took coordinates and then you subtract off program I bet you will remember that but the point is there is always an energy it's always given by a certain construction you could go back in your notes later and check that this is the definition of energy and it is always consider its concerned as long as the LeBron James does not explicitly depend on time no explicit time-dependent and so the energy is considers all you have to do now it is taking wears peak years PR multiplied by Aug . 2 P ALP say there is just multiply it by thing about it than together and subtract Lagrangian sounds
like O'Hara but it's not so bad what you get is pretty simple H is script F O R 4 times the mass provided by girls square root we will not need PR any further only needed to calculate the energy and this is of course is equal to the energy and it is concerned does not change with time but sources put on the ACS wears really rare really wanted the energy script death Laura can and then square wrote and now I actually right in what goes into the square root here but are now at the stage now where I'm going to use the fact that we're interested in circular on the circular orbits are about equal to 0 so for a circular orbit the energy is F minus no no miners G are about nor about minus are squared fate about square but so this is an expression for the energy that involves paid about squared the the it's a Barkley kinetic energy partly Connecticut Energy due to the most motion say is not Connecticut G due to the motion of our figures are is not changing for circular No are velocity just a velocity This is a combination of potential energy are dependent and it has stated that has velocity dependents and its conserve Bessie energy what about the angular momentum he added the angular momentum is also considerate and has the same denominated have minds Our squared they've got square look at what you do it there's not tell you what to do with a play with the prescription is the 1st thing you do is you solve this equation related to it at home you solve this equation . and you find faded stock as a function of all our and this is usually called the reduced angular momentum and angular momentum divided by the Mets you find the angular velocity as a function of angular search if testifying angular velocity as a function of the reduced angular momentum and that's not hard to do but there's no point in doing it it's easy do I won't do it but they you when you get it you plug it back into the sequence into the energy equation and the result is that you have the energy as a function of aura and the reduced angular momentum so I'll tell you it can get says it's always consists of this alternation between principles and having fun figuring out the principles on which you should do and the the boring work of doing little calculations in the back the principles again and I will short cut the boring calculational part we solve the a faded last year and then we plugged into the energy and I'll write down what we get up energy is equal to the square over of half times above the square root and square root is are squared Glanzer squared reduced angular momentum squared plus the old abided by a bar squared up friends and fans and always have and that's it that's it does nothing go lose nothing to it interesting about it how do you find a circular orbit what you do with is this does not depend on Aug . with Nell interested only on the radio the radial aspects of emotion we want to know what radius the equilibrium position for the year before the circular orbit is the equilibrium position depends on the angular momentum for a lawyer named took a good for an ordinary Newtonian problem His angular momentum bigger or smaller further away you go for Norbert as obvious as obvious and reliever to work out whether vigorous smaller further away that industry the further away you go slow the angular frequency but remember and the small velocity but remember that when you construct angular momentum after multiply by all are velocity times are likely which way it got know about hydrogen Avenue cadets but there is a gap probably would be about Glad are squared yeah was it would certainly be but the believe it that way anyway a million for the moment because worth than Eleanor go to the limits of photons and into city now in the bizarre question of whether a photon a bit of the year of the geometry of photon ever getting close enough that the speed of light but that is not enough to ejected from the yellow from the circular orbit so we up for a circular orbit and the rule is to minimize the energy as a function of all are for fixed angular momentum OK but now this is this something going on but times have 0 man so you have a bit of a funny problem he photons have 0 man s back to remember the angular momentum was but tons of 0 mass so it looks like they have 0 energy but they don't have 0 energy a moving photo a high energy photon marketing out of the hits you why because it has momentum and energy OK so photons delay energy as an goes to 0 0 something had better get big cancel out and they also have angular momentum for all momentum momentum and angular momentum of photon efforts moving around In order for use just going past the does have angular momentum so what happens is goes to 0 the answer is that the thing that you hold fixed busy angular momentum if you're interested or photon of a given angular momentum you hold angular momentum fixed you let the Mets got 2 0 but obviously that's gonna make land the incident Miller's alright later take the limit for studying 80 full-time our fixed energy and I've fixed angular momentum is to take the limits of a man's Costa 0 land becomes interfered in such a way that the product of In times Lander stays fixed angular momentum stays fixed now let's just look at the formula what does happen when when they get very big another way to say it is a land that is the angular momentum per unit mass the Mets goes
0 0 lender will get big if we keep the angular momentum effects is an expression land that is be the biggest thing sight and his those 0 so let's allies a if when that is going to infinity Abe value of all our land the square times are squared would be eventually bigger than are too low for a much bigger eventually are squared plans land the squared will be bigger than on the 4th and much bigger so we can simplify this and just say what's on the square root it is just squared land squares and forget the the 4th because negligible when landed at 3rd base it I was just items lender write such a fancy expression is just odd times Lander friends him but when the time zone is just angular momentum and so what's in the numerator here what's in the square root any case times in the square root times and is just the radial distance times the angular momentum is the limit in the limit that would passed all or missiles particle holding the angular momentum fix this just becomes land that kind M which is All Times Square our square and we have this thing over here this thing square of Africa put that back in a minute figure our batteries and finally we have all won all our squared downstairs which means altogether 1 of our 1 0 R L and that's the energy what was in store a calculating with calculating the the energy of each of the orbit it's a function of only is contained only contains Ehlers a multiplicative factor the interesting stuff as a function of all are it's a functional are nowhere His equilibrium where is the equilibrium equilibrium is when the energy is stationary you could think of this as a kind of potential energy doesn't depend on any velocities all velocities of the eliminated out of this problem the radial velocities been eliminated out because it has no radial velocity the angular velocity was eliminated by substituting for the um by solving for the angular velocity in terms of the angular momentum have the angular momentum and we have a function of all where is the orbit of the orbit the photon orbit will be where the function of all art is stationary maximum or minimum subtle if defined as defined where I was valuable are a combination square root of F along with achieves either a maximum or minimum value OK let's now go back remember what ever martyrs advises the Schwab still factor here just for GOR strong square root of 1 to MG or are all divided by R let's imagine plotting misfortune but urgent plowing the function and trying to see what the minimum what what looks like its minimum wages and there but that sorry is not a minimum as you will see on a minimum OK 1st go very far away will you go very far away to NGO bar is much smaller than 1 negligible compared to 1 so you very far away this is just approximately 1 overhaul another words it just falls off like 1 but why happens as you move inward as you move inward moving into William only in Word Cup are at the center of the origin here is now Oracle 0 and our equals 2 energy the horizon Boeing toward the horizon end at our equals to what happens to the numerator he becomes Iraq denominator is not 0 effect point so the whole energy function becomes 0 at the horizon so it must look like this and that is where it looks like now how do you find the point at which I which it maximum value find the point at which is maximum you differentiate this thing a new set of equal to 0 I will tell you the answer the answer is a notice city of notice politely that will I left out it but elders multiplies today's being the properties of a maximum or minimum or whatever it is will not depend on L OK so here we are differentiate the square root divided by are said equal to 0 Where is it right over here and that's the point are equals the rain 3 G there's a maximum to be energy that don't potential energy that has a maximum does that correspond to equilibrium position yet it corresponds to an equilibrium position but stable equilibrium and unstable equilibrium means if you put them object right at equilibrium position it will stay there if give it the slightest half of all are the small the tap you give it on the smaller the imprecision and putting it right up the can't longer will last at the top of the cop but eventually to will roll down pick up some speed and fall off the top we're talking about a circular orbit we're talking about a circular orbit of a massless particle we stand and 0 so what we found is right at articles 3 G for massless particle of photon can Blackall it's not destroy chill radius it's 1 half time too short to read that radius is called follows the euro sometimes a photon beer obviously reason it's a sphere from which a photon order are after you might expect that if you had a black called anew went you look at that distance you would find all kinds of photons orbiting around it has drawn the ire any photon they did start in that orbit will stay in that but of course on if the photon starts slightly away from the copy adjust This is a test of which starts out a little bit too far to go back out that corresponds to the or bits of photons which are Out Beyond the photons fear I wanted just a orbits which get cold they looked like hit the photons fear in
1 of the 4 times for you I want to be outside the photons outside the photons but just outside the 4 times the photon will go with around in income our not necessarily exactly the opposite direction the closer you get the spirit more angle full-time will sweep down before eventually go back down some particular very very close of 4 times a year a full-time coming Gloria many many times but they go back out that corresponds to just be slightly off to the right here and slowly going outward while you could you could imagine coming in not quite getting there and going back out why happens however if the photon was on that side of the photons on that side gift pulled right in it can't withstand the gravitational it's just slightly inward of thief for years instead of 4 buildings may ordered a few times but will spiral right in the horizon and then of course eventually spiral into the singularity OK so that's always see from the spill calculations now they deserve we can't all what deliver I'm interested the photon will be given angular momentum that those From young yeah well and photon has some angular momentum so I might as well equal to what its angular momentum it angular momentum is conserved so whatever it is it will stay that way and Odyssey ordered does not depend on the angular momentum angular momentum factored out of angular momentum factory are the orbit of these B unstable equilibrium orbit is right articles 3 G and it doesn't matter what the angular momentum so we're thinking classic with soap your cups of photo it has a certain material guest deterred by frequency end it video which radius of the city different angular momentum about that said but but not right but it could be now it could be ethics we could be talking about being a definite radius and can have different angular momentum the even angular momentum will correspond some of them will be back some of them won't audits and go off but we could fix both the radius and the angular momentum independently not if we want that circular Robert everyone that circular orbits the radius and the angular momentum are connected but not at the photons feel only circular orbits of circular orbits can have any angular momentum methods curious it seems like you your freedom of fixed 8 of great value do you do you do it will not be the only secular order for the fuck time is only 1 circular Robert we are not here you have the freedom if you award of circular orbits but on the only kind of circular orbit is the 1 that 3 and that's basically what we derive all concert Corvette and unstable circular orbit of anything that's inside any any folk that is moving but say along going on NATO and angular direction that's inside the photons gets swept in by gravity and of course anything that's moving slower than a photon has even more trouble staying outside solve anything which is inside the photons four-year moving in an angular direction whether it's got a man's so it doesn't have a mouse will always want the following year jerk approach now I vote for years but about that where you will find the scribble Ward orbit that as if he analog of saying that there not stable of equilibrium if you looking with talking about a kind of equilibrium why what kind of equilibrium and equilibrium with respect to the radial coordinate so the acquaintance of motion of the radial coordinator fixed angular momentum I have a solution which is just too extreme mobilized the energy to saying operations when you say equilibrium position corresponds to a minimum or maximum of the potential energy potential energy is a part of the energy which doesn't depend on velocity here talking only about the radio motions were talking about a part of energy which doesn't depend on the radio motion does depend on the radial velocity Okayama figure batted well on bit compared with the corresponding the Tony calculation the seal a pieces are the same as in the coming calculations of calculating the the arms but the outcome was quite different outcome quite different to that them particularly on circular orbit anything photons or otherwise there no circular orbits within its inside 4 times fear they are circular orbits from massive things outside for can't we're much of Nestle's particles only the far apart but that's just a young an example example of working out a specific most of our in misfortune geometry of this what it would look like a plastered on their are you could actually where are we looking at looking at the black the stench facility a black coffin it is a buckle you're at looking at a bicycle light that passes out here Men's a little bit so you think about what you see in the backdrop figure but the backdrop were behind the Blackwater see you don't see the black black all the medicinal light our so what you see is a backdrop the stars or whatever it is a Picasso painting behind ordered is you say it the costs now would you say it is it appears that everything in the paint thing shifted your luck and you see this point as if it were as if the light were coming from here moving get shift out a little bit away from the center but it gets worse than that as you look in for example will be our alike race but goes from here to here and other libraries across from here they have maybe a White River goes from there the and you'll see this point double but it isn't just point doubled is pointed to see double some point along here you will see double are in fact candidacy of WC Split half used to mask its no not polite breaking of this way wrong this
way they can go this way it can go behind the so would you actually Caesar a kind ring around the block or stirring Blackall and light raised 6 angle around that can make Ken Starr over here and go like that a black hole is a very bad thing to try the hide from your enemies will not called for a hard to hide from you and me by hiding behind the black hole of enemy is over here and your over here you may think you're hiding but you're not because there are light rays which like that but of course what you see is very very complicated as you line sight moves closer and closer to the horizon because you'll see the light rays which go around many many times and it will fragment the movie you have the background and were just and impossibly complicated things now Einstein rings are around our experiment with observational verified things said that both on their right to vote while I was thinking of the horizon over here are light Ray which is pointed outward can get out from just above the horizon the ones which appointing tangentially this way which will fall in if they're inside a light gray around here is that is a year of photons theater here a library which is moving in the angular direction will not be able to escape from the photons fear but library which is radially outward most particularly light rays which partly radially partly angular and those will swaying always around America for example he right Sarette makes armored looking your black hole is a fine Seubert complicated pattern of more fired cat and should worry about the virtually no work for federal case shattered the eyes that I'd never done it that's a match it so far up your modern computers just so the equation on Burma computer that rules about answers right Back that it Orza said where branch that gave orders for young players would or not it's not Eds as follow led probably takes more steps in the right how would you do I suppose what you would do that I'm not saying I realize had not the question on your right hand or brands equations and then you stare at them and say how the devil all I use the conservation laws sometimes I have enough conservation laws that a completely determines the answer they you don't have to solve the differential equations and that's all we really did he I could've written down medium the equations of motion and observed that there was some conserved quantities just use the trick of starting with the actions always the most efficient way to do things about as it always has I think it is always may be the case here a mere whether something her easier but usually OK now we want to understand better what it is it's happening armed at this curious point it's not a point as Kyra shell around the black or straw again he is Oracle 0 but steadily likely Horizons a red horizons Red under 2 so as a kind of shallow and the black hole the horizon and something curious is going on among other things the signing of the coefficient of DT squared changes as the of the coefficient of Dr square they both change they keep thinks the number of negative signs and positive signs but still something curious going on we know that clerks when viewed from faraway Clark falling in the seems to go slower and slower bets that relationship between in town something is happening here on the other hand have also told you that In some since nothing is happening so high we reconcile nothing was something the fraud she many things that happened in gravity to be understood that 1st understanding them for accelerated coordinator that's equivalent principal are said gravity is not really exactly the same as a uniformly accelerated cooling system for its a very good thing receive can see what's going on for a uniformly accelerated port system so let's go back to that problem the problem your uniform acceleration in relativity is a little more complicated than it is end rolled plutonium physics Blu-ray talk about it but corresponds Coke whatnot system which is the analog of slum polar coordinates but polar coordinates in hyperbolic geometry for approval of hyperbolic geometry just means circles are replaced by hyperbole focus on just very quickly if remind you that I introduced the flat space now plain old flat a black coal it's just ordinary flat space-time are there may be some coordinates coming out of the blackboard for the moment but that would be interested in them but I am interested in is scored talk all capital X and discordant which I will call capital time and space and then maybe 2 more coordinates which you could think of was coming out of the blackboard but an actor playing a big role in world a safer moment OK that was the metric of this show is over next we introduce hyperbolic coordinates will reintroduce polar coordinates of the right feel definition is that X is equal to something which our call role roll it is distance away from this point over here horizontally rolled hyperbolic call sign of all may get I maybe it is
a kind of kind variable that varies along a hyperbola along each had blood all maker varies it's a kind of angle but it's not really an angle stop periodic it's a hyperbolic it goes from minus infinity down he had the pluses and Pierre but this is the analog our X Eagles role cosigned data for ordinary would Kohler Court where about C is equal to roll hyperbolic signed a lawmaker what about X squared minus he squared x squared minus D squared it is a quota rose squared times cost squared minus send square cars where my usage quit 1 so each 1 of these hyperbole is a trajectory on which X where miners T-square constant and that is what we call a uniformly accelerated coordinates and special road through special role at the Met correct the deck of ordinary flat space is given by D. pal squared equals rose squared deal makers squared minus D rose squared and that's see an alarm Our squared the status quo and plus Dr squared would be metric would be with continue cholera square Square dictator squared plus the euro squared would be the metric and polo coordinates for an ordinary geometry here it is for around our Kowsky space a special road cows Cuevas rose quit the Omega square for minors Euro square nonetheless change sunk coordinates just boring very very boring chord changes from our rent from Rome that the called XCI hate and I hate making new location but we have no real choice we have to uh we have to study this theory in different coordinates with changed course that's a little bit of a cake rose squared here call at 4 times the Greek letter seat Cseke rose squared is 4 times exceed I'm going to call or write that down the road squared physical before times similarly former similarly but also were going to write that all God is just see over no big deal there it's just convenient to work to replace Omega like tea over to now re Though the magic the Omega square with steelmakers squared the Omega square is squared over 4 cases this is Dedie squared for that deal maker squared DT squared over for and now we have the multiply that by Roh square cut above the rest the year and that's just 4 times exceed Biggio c times DT squared at OK now what about minus D Rose square with sick mind Bureau squared his squares calculate D row square differential of Rome square but differentiate this equation to roll Bureau is a quota for big and means that the euro is equal to I think it's 2 0 Dixie overall driver right moved Madeira Chip OK so bureaus squares is Dick C squared all over Rose squared but now squares Foxy put that here was canceled now maybe yes maybe no you will notice of similarity with this watch over record and the short don't let you have a coefficient function here times deeply squared and you have the inverse coefficients function here times DR square do you think of C is being like all are this test some family resemblance to various notice another thing back this change assigned when I gets until clogs changes starring when it becomes negative the succeed being negative means something yes we will see the exceed being negative does mean something so there is a similarity of the basic similarities is a coefficient another coefficient which are in verses of each other which we will seek and be fully change signed as see becomes negative OK let's a flesh this out a little more with less is more what's that now they wear the back foot the batter but for the moment we just concentrating on the radial and time like direction and they will put back the other 1 also but for the moment was just concentrate on these 2 guys over here OK is what I wanted to I want focuses resume zooming in on the placed or equals to a blowout focus in on all our which is just another way of saying we're the study this magic in the region where authorities almost equal to toe MG justice or blow up and see what's going on there figure microscope blowout OK refers right at this way are minus to MG are I remember anything just a move the numerator over a month later now I have to be K-lyte were only in the going to be interested a variety is the horizon and I'm interested in the vicinity very very close to the her eyes I want to explore that region very close to it which means I'm not going to let very
much but I know are change very much you would think I could just Senate equal to tow energy that I have a bit of a problem of enumerated here I would have 0 I don't want to do it With the numerator but in the denominator the denominator does not change much it doesn't become 0 Oracle's tool in G nothing special happens to all our when it becomes equal to tow energy implication is that an approximation of where you just hovered near the horizon Yukos said to 20 without losing any any accuracy very very close to the horizon that's like in Newton's equations Newton's gravitational theory if near the surface of the earth for many purposes you could set are equal to the radius of the earth Will the careful about things which go 0 and also things which affinity and this is something which would be obvious look businesses are minus 2 MG squared and in the interests to energy are minors to energy br squared noted that just simplify this I wanna carry along too much baggage I'm going to set a Of the black hole to be a specific number on equal pulling number so that to and jeez is just equal to 1 that will simplify a larger lot here I can always do that that's a that that's a choice of Unix was the units of toe injury you're too late to energy is the total radius as units of for a particular black hole I can chew loose units of Lane where are we sure to radius is just equaled the 1 I can't do it for all black calls simultaneously but from interest to a particular black all or I can do that this formula much simpler let's just do that let's set to Angie Cooper 1 its are minus warranty he squared 1 of our minus 1 square Dan but what's annexed out but the next step is to redefine or minus 1 in call exceed just a change of variables it's a change of variables which measures all are relative to the horizon now the horizon is articles 1 articles 1 because so what would ever do is with change again so that I no coordinators just measuring distance from the horizon just measuring the deviation of all are from its rise in value let's call are minus warned exceed this 1 becomes 1 although sea they are squared but they are C of the same thing with 2 or the definition exceed his Eagle bar minus 1 deep-sea is just equal to Dr service becomes Dixie square exactly what not yes yes indeed exactly what we have here what's going on what this is telling us is that the vicinity of the horizon is in some mysterious way just flat space or approximately flat space for nothing nothing dramatic is happening because by according to transformation if by coordinate transformation I can change the Mecca so that it looks like this saying that to an approximation of good approximation the vicinity of the horizon is locally flat nothing very dramatic going on a large curvature more or less flat face that's surprises but stroke call for flat from you it to the uniformly accelerated Accord yes but that doesn't mean the space is flat but it's flat space in which there is a coordinate system which is moving with uniform acceleration while would really be interested in the coordinate system from gravitating object which is leaking uniform acceleration well standing here I am experiencing an acceleration of 10 meters per 2nd per 2nd I feel exactly the same things I would if I were in free space on the floor on the knee was accelerating upward with 10 meters per 2nd per 2nd In studying a gravitational gravitating object from the the perspective of somebody standing still being supported let's say above the horizon or murders your effectively doing physics uniformly accelerated reference where what the physics feel like they always do that it does whatever whatever uniformly accelerated reference frame and he we see little it would just seen exactly I was seeing the vicinity of the horizon of black hole is simply a uniformly accelerated Cornet system but the space itself the time itself is effectively flat not much different in that hardly different now that's a a the forward let's talk about this in version of space to Time and time to space but was Eagle minus 1 floor if I put back MGU would be related are minus 2 energy I sipped orange equaled 1 but is a truly proportional Barmer just seeking change sign as a change signed by going from outside the horizon inside eyes so was before yes it makes sense at least the black hole context here to say that see chickens can change sign doesn't make any sense in the flat space and if so what that where there is negative see on this diagram so and so that we go back they'll look here's what role for his sexy will succeed was equal to roast square all water the 4 is not so interesting but a point here to get for some port cheaters rose squares off for doesn't
mean thing referred real square to change sign Alcan rose quick-change Cyrus squared as road was the positive numbers square rose some real numbered squares positive not that's not quite right the way you think about it role is the row squared off on the right he inside of this equation positive rose square here means are hyperbola the In this quandary hyperbola In this quantity over here as a positive Rose scored it means that X squares be given squares X squared is certainly big T-square all along in in this entire acquiring X magnitude of X is bigger than the magnitude of in this whole quadrant here that's why rose squared positive was Ross would be negative me it means a T squares is bigger than X squared up here let's forget the bottom of the left so was just concentrate on year on the squad squadron for a moment Rose who have been negative simply corresponds to a point and the upper quadrant that's what means rose grabbing soul in going from seat consulates exceed negative is simply passing from here to here a man to know that we wanted tools follow our way unit now the file our in till we got Trixie equals 0 Buck and they will only go from their visit with doing on yacht here is made eagle 0 and where following a role were following grow into the origin here at which point exceed changes where do we go from here as following maybe equals 0 end wedding vary we Uncle here we look here the we jumped into the soccer maybe equal 0 along here no make equals 0 he he see is greater than 0 here city it is less than 0 cell it was happened is yes a space like coordinated namely see has jumped and become a time like caught it from a guy which was angular corn hear what happens to it what happens if we very maybe in this quadrant we're moving along here also I privilege it was at light yet but nothing is just that yes but they coordinates wouldn't know anything you want no physical system is moving faster than the speed of light weather system is moving fast speed of light is quite separate from whether a quarter areas is moving restrooms pure white so this it is funny coordinator it is a funny coordinates where goes are not rose salary would exceed Brixey goes from being a space coordinate the being a current and Omega goes from being kind like inherent to being space like in here however remember all this was just flat space was nothing going on that somebody who jumps here sees nothing special nothing happening it's just peculiar coordinates have some funny behaviors you go from 1 side to another the same is true of the caught C is Arlyne minus 1 are minors toe injury when this change assigned when you go from here to here you doing exactly the same thing you going from here to look at but what goal on a little bit and think about black hole seals are minus 1 very far means varying are until you get to 1 that's the horizon To energy and then continuing very are you move up here that's the nature of these quarters but eventually you'll get Oracle 0 hero Aura's decreasing decrease industries decreasing Tiller gets a toe injury and then a key is decreasing until you get To some point hours equal to 0 ah is equal to 0 something nasty does happen when order is equal to 0 something else is happening where is Oracle 0 Oracle 0 Over here you're behind the horizon inside period to the horizon the change signed guess what we singularity at Oracle 0 is not a place at a time a not place at a time not single place here may different places all packed the same all active well the shortest same time but may surface equals 0 is not what it thought too was a vertical line like this it was more horizontal and more space like the kind of as Oracle 0 but because using right 5th answers confusing but there are brought this away the black all really you get used to it will be used to think about and play with it the habit we don't you Dad for a little over when it for that short for days by death debt at 1st never proved that or that but physically what happens for system that you can use it you could go you me I don't make it a mark worried about maybe you'd be appeal think that maybe you'd be appear but the problem is you have to go through this is visit a sorry 0 still was like is what it is that you could think of was a center of black hole Omri bath obviously it's a funny kind of sent there on him as a kind of place if you
like that's why you can avoid it you can't avoid the future with no way to escape from the future a place thing is a place I can avoid like go around it why can't the avoid is the future that's why you can't avoid the singularity when you get in here because no matter what you do it's always in your future and you will hit just like the clock wound down to evict sorrow the singularity is not avoidable the way it will it would be in the timely in physics in which the sector of Our coordinates is a place you go around the place you Kinkel around the time so that we are that's that's the nature of the black hole singularity What about the horizon while the horizon over here but think about a little bit differently eventually would think about this whole line is being horizon but but slips erased some of this yacht that Air 1st those who act like it the system is not Ross said there are data text were were those that was a lawyer who Michael Drive worthless Our a W mine is yet are a particle might start over here come in the past I know this figure explain something tourist this figure allows us to understand much better this idea that it takes an infinite amount of time for think the fault of the horizon at least in incident amount of Oakwood NYTimes but still it falls through it only timekeeping if also a finite time so let's rejoice Suva commands back but at bay P same picture enallage drawn what time slices look like these are different times all makers like what was a relation between them with a yacht in comparing this Over here what was a relations T C was May over tool comedy Omega steel right so time is simply this angular court position is constant along with hyperbole is now let's imagine somebody Alice throws Bob into the black for Alice stays outside was in Alice has coordinates which are just these farmed poll look type coordinates Omega where his or eagle 0 make equals 1 of his own made equal to his armored vehicles is on making equals for and OK a very good for Omega asymptotically goes to infinity when you get out of this like like direction here so Alice she says infinitely Lake time corresponds to the surface over here and fact if she's looking back and watching Bob she looks back she looks back she looks back and it's not until end In an amount of her time has transpired that she actually sees Bob go through the horizon another which she never sees bungled the horizon call Idaho reckoning that takes an infinite amount of Omega before Bob Walsh who arrives at translates to an infinite amount of teeth in these coordinates on the other hand just looking at the diagram you concede immediately that the amount the proper time between here and here is finite just an ordinary trajectory nothing special happening to it it's ticking off time a finite amount of time in order for the subject of fall through her eyes this fall through this surface here is the horizon but will come back to combat so it's a quarter property of funny coordinate property that Alice's reckoning is that Bob Dickson in the amount of time to fulfill the horizon not so says Bob Lohr took a finite amount of time I watched my clock but the reason for the discrepancy is this discrepancy between proper time and coordinate time Alice uses Quark NYTimes Bob uses proper Wheat young Alice's staying outside the black hole she's like I am now undergoing uniform acceleration she standing on the floor above the year above the horizon black hole and so she's more like this if you actually go watch what's it but just imagine Peru looking but she looks back and that this incident over here with Mrs. C Bob she sees Bob back over here Barbarea Lewis just look back in time simply because 6 time like some plan to get will do later she sees Bob over here a little bit later she sees bobbled here she could wait forever and she will not see Bob course outline Bob crosses the line as far as he's concerned it doesn't see anything peculiar me a lot of traffic OK that's let us look at work singer Bob before I answer that let's just say 1 more thing Alice obviously sees Bob slowed down while the slowdown because it takes an infinite amount of time foods are very large amount of time from her perspective to go from here today at the from here to there it takes a huge amount of time each heart beat of Bob takes longer and longer from her point of view so how also sees Bob slowed down what the biopsy settlers Bob was Bob see right so that is Bob that is Alice and Bob looks back he sees us over here he looks back issues Alice over here he looks back right at this point worry crosses this is nothing
special even a after he passes through 0 Alaskan never see Bob at that point but Bob has no trouble seeing Alice so about sees nothing specially Sialis speed up doesn't a Sialis shrinks we is a dusty Alice accelerating away from him that's true he sees Alice accelerating away from him but that's always our served at any point he sees Alice uh are perfectly normal except accelerating away from him you know is she's willing with uniform acceleration as because she standing on the floor to standing in floor at a fixed our members doing have run she could be ordered she could be in orbit young got what she did just be standing on a platform that built above the black hole Majamay platform arm or suspended suspended from a very very distant object whatever waste is required to keep her just at fixed height above via Michael affixed high height above the black hole corresponds a trajectory right back Bob looks back nothing special happens talents when Bob crosses his point on the other hand some Alice's perspective she never sees Bob Crow said . 2 takes forever from our perspective so it's very asymmetric music the slaughter and while the singularity right so up here is a singularity someplace and Oracle's oracle 0 Bob it's a singularity right over here by that point lead tidal forces will become so large the Bob will no longer be with us marble destroyed by the time we get to hear that's why we don't think about trying to answer the question with Bob when he gets what lexicon negative are about country it's a negative are he's experienced infinitely stronger tidal forces and there is Bob all I know is what kinds Kyle How was proper time but a Bob measures along here have a parallel to all wall in here inside yacht these society in this quadrant here Bob would probably use all are 4 times or something like our foot yacht inside him as varying from here to here from from Oracle's toe injury he rhetorical 0 here so his clerks would have more to do with Oregon T and T be more like position in here but has said he doesn't feel funny thing happening waste favors Clark's stock becomes a muse thickens meterstick becomes a clock nothing like that sales through peppery now honey gifts the parent that this trying already yeah is forever special is rather special is that anything that crosses behind hear anything across behind here can not escape will eventually remember In the coordinates that we using light woes were 45 degree angles like those were 45 degree angles these coordinates and so polite cannot escape Our here cannot escape back to hear all I can do is eventually had a singular in anything that moving slower than the speed of light will also hit the singularity so anybody who finds himself In this quadrant in here is no some money out here has the possibility of escaping put such talk about libraries of a library is moving radially outward here it will escape simply just keep going Interscope pointing radially inward it of course will love will also do that with the singularity is everything in between traffic light raised for how would it may fall in depending on whether it's beyond the photon radius or not but this is a picture that You wanna get used to if you want to understand your be able to resolve funny paradoxes about who sees 1 the black all go back to the picture remember that Alice's time on the outside Alice was far away or movie with uniform acceleration Her clerks recorded this hyperbolic angle as time bombs Clark's recorded proper time that mismatch that mismatch which is governing all of the peculiarities of emotion near Blackall OK I think Gloria through a young black hole for me so would have thought that from our perspective such work is still so similar trapped inside little absurd of much talk about what our seas stays outside the black Alice there we we are speaking about classical black holes were not going quantum mechanics Alice seas all of the matter forms black or not falling into the horizon but kind sedimentary motion with everything just asymptotically getting closer and closer to horizon in close Close course flattening out and of forming a kind of statement to structural layered structure which just asymptotically migrates closer and closer to the horizon that's a matter of former by call her for me here by all here that means that new burst a lot of it every day at work but it is not OK so How was jeans changes at hand I'm not sure what happened Charlie I for Charlie firms and frolic comes
late In order for the state so 1st statement is to see anything like has to come from the Charlie concede Bob is because Bob a legal right now arm Charlie sees Bob's clocks slowed down privacy concerns all of mobs clocks moving slower and slower and slower but that also means that Bob's light sources are he meeting with suppose for the moment that Bob sending out polls signals 1 single from amount per microsecond and what Charlie is at 1st sees signals coming out roughly at 1 per microsecond will later it's 0 1 for 2nd and somewhat later so repens seconds stretched out more and more and more why is that this is a game that raise your factors end errors Bob gets closer and closer to the horizon is a big biggest stretch between them and neighboring pulse if we translate that the frequency of light it saying that the frequency of white is getting the frequency of light emitted by Bob Bob Bob thinks he has a flashlight that sends a light that that a particular frequency he thinks frequencies were tend to above about cycles per 2nd but the light is registered as far as Alice's concerned the 1st Charlie concerned light is getting more and more and more progressively redshift so workers Bob sends out conventional optical light will arrive at Alistair Charlie 1st as conventional light man was infrared microwaves and his radio waves and longer and longer radio waves of course but given number of photons and radio waves as a weaker and weaker signal and less energy so the fact is that Charlie after a long long period of time simply will not be a practical way Bob Hill Bob's light having been shifted into the very very very far radio which becomes harder and harder to detect so the answer is that some of that Charlie will see nothing but it is capable of keeping track of these very very faint long wavelength radio waves he will deceive Bob slowed down you they are like the singularity he said it had no more and no less than the picture no more of our whether we lose Bizet every trajectory for here that is woven with slower than the speed of light will hit I will get to leave it at at some point I that's what that's all it means is that that any trajectory starting from here cannot avoid it and will eventually hit it in some specified amount of time along with trajectory that's all set but wound 2 1 on a visit technical it says that the surfaces space like surfaces space like all directions within the surface within the surface or those directions of space flight and there is no kind correction within group net surfers if backers have a go at but about ethnic also share plus a figure of an informal yet as a massive object whoa saw the horizon and the horizon comes out to meet it How will thank them for that you have to solve Einstein's equations but in fact that's what happens the horizon comes out the lead it makes it at some point it blends in with the horizon for a it and made his bulging knowledge in the horizon spreads over the whole thing and increases the size of it lot still wear the point of this thing coming in here always being on the outside is firstly head this set a huge nets in extremely large debts in the dust is slowly collapsing the best the best but yet but at some point it would reach that that's became so it seems like from our perspective there is gotta vast inside the black good although which was somebody watching somebody watching it from the outside from the outside will not see any the rest passed through the rise nobody was there before their eyes and started so that its big mess little Wallace does but I did the best accumulates to undergo the gravitational collapse you get support rate set there but we see we noted that the center so how do we know where the problem of creating a black coal him by Indian falling shells material to see exactly what happens with the horizon but may not that rather get had every right now I think next time we will work out the picture that goes With the falling show of matter no black coal to start when the falling show passes its Schwartz radius there will be a black hole the horizon actually begins before that the horizon begins in the center grows out of meet in falling shell what happens now but they article 0 with no black somebody throws in a in falling shell matter we come in court that shell matter forward job matters falling and guest article 0 right over there but
before it gets so the origin add up at some earlier time horizon starts growing from here and that horizon or something like that Miller Lite can ever escape from inside the horizon outside because to do so would have to go faster than the speed of light so that means Alice looking back I will see a light gray coming from here from here from here from here to tracks always back it will come from here you will never see her she will never see this Shell passing through the horizon over here to look back and trophy show at 2 years later latest look back and see that Shell getting closer and closer glamorize and the passing it do that problem with the problem and detail for too ability equations just the church for 88 whose use a higher first-half becoming from where bigger Media light did not revealed these days but too M and Ferdie GM yes vote a look Be that life the source of the life yes but if we see the stars aligned in yes a gang I we seals of Hoosiers also like between Fujian and 3 and Moreau we see a source of light we can get more shit like get inside Fijian know it can escape this moving radially outward now is to educate his 2 Jeanne horizon his Have we discover a light rail he happens to be moving but say in that direction we can be sure that a fall that we a light ray in that direction no problem right lightly will skate of which Eli array of that direction we know so any given the yard any given distance you could divide the different directions into those which fall in and those which don't fall and if you are at this point here and you consider a king Genscher library in other words 1 movie with fixed radial distance that 1 will fall and not so if it's out beyond the photons fear it's tapping photons year more about what's inside for France fear fall in but as long as Iraq side to GM radially outgoing library will get out of that do that late in between that you're right yes about to tell light at Beirut book no no no I asking case but I think you're asking I think used his our eyes and ears of photons here and I think you're asking whether a light raking going to come back out now because of winning came back out the would-be some point at which it was moving tangentially wanted willing tangentially ill will fall and so not a library cannot come and go out of the year before another not property for pranced for years but on the other hand if somebody doesn't have to some somebody's some poor on happy person who was fallen through his motion is such that he knows it is doomed because still send out a light gray radially outward there will get out not so fully passes the horizon press our eyes and no way to communicate with the outside the only limitation issues there you will have to send white rate pretty close to radially out at least not to work up to shallow here now said site but I right about now but said but is not our well that's an interesting question from dropping the actor get there a little bit this bit inside suspended it's just a little bit inside a light rain which is just a little bit inside which is moving to the tangential direction will go around many kinds before so yes they can communicate now OK focuses an interesting question which I'm not exactly sure the answers supposing they were diametrically opposed positions at some all are what are the limits on whether they communicate I I suspect there must be a distance beyond which they can't communicate our work that out of the hard work are fairly and tour of him but if provoked as long as is sufficiently close to his point he Yardley can but even that gave it send out weighty which would then Colonel job that's correct we could have remember could have removed Charlie out here Alaskans send the message to Charlie Charlie can take that message born around year perfectly nicely in orbit instant message try of account for In the beer they but actually get tangled where there about that was clearly no opened the limitation on the arm murmur communicating the range if only 1 such like as move closer and closer the Verizon he I think Pickens still communicate the way they communicate his Alice consent about almost radially outward almost directly review our work doesn't have to be a light raid could be a rock she throws rock with enough velocity to almost escape and comes back right now just so they can make can communicate altered could emerge with 4 more please
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