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General Relativity | Lecture 5

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so a lance Steiger University Fry let's
talk about friend like space like em might like that we just go back pressure wrote to be served by whatever means of special relativity meant that we discusses meant that many times we call a proper time Bacau squared because Cuevas where the proper time some little the and as your number he squared minus 3 x square mile y square easy squared everyone the speed of light sometimes up with the speed of light the reason I put the speed of light and sometimes is to keep track of what small what big under certain circumstances for example if you want to study if you want to go the nonrelativistic limit The limit where everything is moving slowly that's got to put back the speed of light because he keeps track in the case of the fact that the speed of light is much bigger than any of the velocity the problem so that put in for the moment I'll put the taken out there are under various circumstances because it like that for overseas squared and if D-Calif squared is greater than 0 50 pal squared 0 now of course the square of a real number was always greater 0 but because quit is not defined by being square real real number it's defined by this combination over here and it can be positive or negative depending on whether the X clip was dealers Clippers was Desi squared is bigger than D squared all small squared off I said DX wept as why script was busy square but I'm in the X squared overseas square your square overseas so full of banana case if 50 he where is greater than 0 mental element here is said to be time like the got more time has space this card as big as part and that can be described in terms of like all like all like Conan books like that but and the boundaries of the cone called
itself of places would be key squared let's go for a moment I don't want to scores that fire BP square he
also DX equipped BY squarely busy square that's called like like D squared is bigger and DX quarter plus to 1 script was busy square it means little vector lies in the interior of a common like that a could also be lying in the backward direction in either case a little interval or vector of speak about veteran general Beckham general time like it's time component is is bigger than the sum of squares of space components of plan Mike space alike is exactly the opposite is deeply squared because less than 0 and that case we usually defined another quantity fate s squared which is just the negative did Bacau square by definition is yet square posted 1 square foot busy square minus squared the same object except for the relative with different glimmering assigned space like vectors all ones for which a the S squared is bigger than 0 because we're lesser 0 in space flight vectors lie
outside White Kong finally like like vectors the space flight and finally there are like like that there's and like like sectors on the ones for which indeed he
squared equals the X grapples grip was those are the trajectories of light rays and they lie in the may lie in the surface of the itself 3 kinds of vectors you can have on just from moment consider what would mean if they they want to negatives sorry to positive signs into negative signs that 1 positive sign and 3 negatives somehow this would correspond full time direction the that doesn't mean any anything thing experiment the 2 times the always 1 time and free space now could you on imagine a world with took I can't frankly I can't I can't imagine will mean that 2 different forms directions will simply take the view that the facts not option there's always 1 time like direction in the metric here and 3 states like but that doesn't mean that there's a unique direction which is kind like them any time like directions point with the slight come here the invariant statement is a statement about the Met cancer the metric tensor weaker write this summer form be pal squared is equal G communal where a minus sign VX mu VX know me might signed is a convention always defined so they're Gino TX new VAX goes With these bases like vectors having a positive sign incidentally 1 other thing it does remind you on proper time all along the trajectory like is a measure of 1 o'clock along the trajectory kicks off proper time is a kind of its kind red off by moving Clark the proper distance proper distance along a space like surface is distance measured by meterstick so proper distance square of the square root beer square really is a distant and proper time really is a kind of and keep them nite OK out of 4 this is our definition of the metric and for the case the we've written the metric To be written in France a matrix North Sea the matrix is minus 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 2 0 1 called a them you know like chronic Delta except that has a mind OK this thing has obviously 3 positive eigenvalue The Matrix 3 positive eigenvalue more negative argued that you and that's the invariant story wraps the story not that there's only 1 kind direction but that here 1 negative I give value yeah and they met trick which would have to negative Eigen values what really negative I mean values would have more than 1 time we just don't even think about that that there was something of a physics the Nazi toward me use 2 negative OK so bad so space flight time like like so but bird more generally it could be applied but certainly in curved space per space time the metric tensor GE the metric tensor cannot be chosen to be equal to the 8 a symbol here it's a general function genial or acts Virginia law G. Mueller Mexican not just be old man and every point that every acts as a matrix of every acts as a matrix of reporting and every X B there must be 1 negative Oregon value and 3 parts of and you are standing there you should experience a world with one-time direction we space or more exactly 1 negative eigenvalue to pass OK sure what that means what that means is is that every point in space has a light cone associated with that not enjoying it sort of had been killed bid that reflects the fact that the Cold that's themselves might be we need what so as you move around because the coordinates and also the metric tensor might be varying from place to place these like cone might have an apparent tendency to uh Oh different in different places but that is just because because as you move around the base of that kind of Cheney from place to place and accord change from place great fracture every point all I call there's shame like space of time like me crime like inside I Kong space like backers outside the light cone and like like promotion of white raise longer enjoyed the call itself the Collins surface when you have ice cream cones uh ice cream and inside is not part of the Hong Kong resurfaced self shelf so I have to make sure that we will only right that does have the property of the right number
of positive and negative barriers call that the property of having a certain number of eigenvalues past a certain number negative is called the signature of the wants as the signature of the method was a signature of the metric of would ordinary flat space the blackboard based Our blackboard has 2 dimensions and the signature of the blackboard plus plus the X squared plus wars the signature or Minkowski space special relativity is but Miners plus plus minus plus minds plus plus plus Excuse me my fault OK so we have to make sure that somebody a somebody gives a symmetric wherever we got from move we might get it present a mail right there we might go calculated from equations of motion some field equations always we should make sure that that Have signature miners plus plus plus and doesn't mean we did some wrong not only to have signature every poor and that at some point it should have every point not all the but there is a single I get eigenvectors so soon may I right now on the no no numbers there now there is no of is not a single was promising or anything anything in the interior of the light columns equivalent everything else by Larine's transformation Yukon Lawrence transforms vector of the not this now you see that supposing I gave you the metric plus plus plus plus resent look like it has only 1 positive start as I look like 3 positive eigenvectors and everything else is not remind you that there not any vector is the unit matrix any vector is that vector were plus 1 here it's true that you work this out any time like vector is an eigenvectors of this matrix whereby value minus 1 a space like back there is an eigenvectors this rhetoric with my plus 1 in any light like the terms of the deal Madrid from the Czech barrels of oil a day at the Shea Asia as well as jail Is the job now that while well large let's let's take this web stakes but let's take this Connolly here this like call with X Y Z T coordinates defined in this way 0 is a 45 degree call with video with the generators of the conga lines of across a 45 degree if I worked on multiplied let's say key squared by some number but Roberts work so provide this here now assistance base it's exactly the same space it just corresponds to changing the x coordinate from X X divided by see a but if I don't show the light cone in the corps wouldn't extant teed it would look like this a C was launched will be squashed is reflecting the fact that like most of the speed of light and is large light goes a long way in those coordinates seems base and that changes based just change the quarter but by changing the court 2nd squashed like all I can even Filpo iconic into all sorts of things close call those like called it does not reflect a real change in the geometry right now since we have just arbitrary coordinates we're dealing with here we could for example have this s feel like depend on position dependent on positions that would change would have it position and metric would have some funny position the and that requires removed from 1 place to another would go from looking this way for looking that ways of pure coordinator issue whether looks 1 way or the other but we invariant statement is that there are space like that this time like that was like like vectors and the boundary between the space like white like vectors call us there is only 1 I 4 fine and in the state want them that not bad it transformed Lorentz transformation transformer Lieurance transformation from the sea now I didn't want to get into it with you you're right you're right but let's look so what's come back with correctly so I get I wanna get through the a metric of a black hole and I in wanna spend time with back of 1 eigenvectors 3 1 9 positive ID value when they tried to rip this metric transforms on the Lorentz transformations in the woods self transformed from the Lorentz transformation into itself for a dual Lorentz transformation we come back exactly the same rhetoric yeah that's the important issue I
told you what the rule for G Odessa growth West time time before it's the cold area derivative tangent vector 0 as you move along most parallel show itself and to give you a different definition for my which in many ways is more useful but money 1st right down the original definition these 2nd ex new bike Powell squares that's the theory of a change in vector DX new L is a derivative of the back Alonso Close ends should be plus some writers think plus Newell said no role the said found VX role that's the definition of overdue best but there's another definition which is many times more useful it's the analog of the definition of a dude desiccant Fortner respects based the shortest distance between 2 points of better yet stationary stationary distance between 2 points stationery distance between 2 points define defines our budget deficit so another way of writing Mr. a week extreme mobilize all we make minimum let's take a case of ordinary space ordinary blackboard for current version of the blackboard we calculated distance between 2 points were interested in the geodesic connecting those 2 points RVG Odessa connecting those 2 points calculated curve figured curve and calculated that was it and dissidents ordinary space would be for each will element of it we have the escalator equals GE and a B X MDX and now talking about space enough space time will come back her space-time moment square or DES is just square that words are factors steal OK that's see that's been dissidents along Lucas geodesic here we just a had not to sit curve and we a them off and they have them all up we just do an integral added that will sign just means a them all out a them all up let the distances Coke Zero yeah them all out and that's equal to the total distance between 2 points the rollover Mr. make that as small as possible to make it extreme all we learned only studied classical mechanics how do we learned only studied classical mechanics this is a problem in the calculus of variations that's a problem and allergist minimizing action for a particle the another words we could think of this as action for some kind of particle moving between hearing here and in extreme realize this will make it a maker of stationery is the rule for calculating that's what kind of equations do we use this Jinyan function of X here of course was the equation called back a lot of those equation that tells you have minimized and like that's called on Lula branch equations will grinders equations for a Lagrangian which is derived from this action think of this as an action particle from here minimizing it is like minimizing the police action when you go from the principle of action to be boiler grunge equations the principle of least action turns In the differential equation typically it turns in the air sequels and type of equation going from here 50 s is exactly that operation is actions or think minimize used a solution it in the form of a differential equation in fact that sort of looks like acceleration is equal to something that something is kind of like OK let's come for the real problem where we're not interested in distance but in proper time improper time if I want to calculate the proper time between 2 points based on points it'll exactly the same thing exactly the same thing except as a minus sign here the Mets people for the proper time between . 1 and . 2 in space 1 . 2 with spacetime marriages that quantity Is it think New maximize it'd be Minamata voting no you minimize the amount that the of loss sorry humor will the sigh right next OK now let's opposes really corresponds to a motion of a particle with started the space time pouring in ends at the space the action depends on want more corn depends on the natural action but the actual action is a quote from might miss the kind the proper current between is definition is a definition of the putting Masson they will find out is important after thinking about energy and so forth but the minus sign it Cleo strictly definition the city where it comes in minus the mayor's here and in fact action I think is typically that minimize minimize the between them Our for particle moving her trajectories really deliver things a completely unrecognisable objects about it probably status of this will not only doing that counted captain
led can deal with that ability and with Darren really there means wasn't me Babruysk with rude of something BX BX what kind of things we usually understand his integrals have a D something 1 would be something of function right some functions the date could be could be time but they're normally we don't see funny things like this with square roots of XTX 5 so this is easy to and wet at the cap what let's break up the interval total times say little time segments Delta team and what's the vise here each 1 of the or but the fight each 1 of these DX is by pretty some of these DXM was might at B B C U S T is or CD 1 of the 4 quarters what happens if we have a deep PDT which Scott 1 1 of those of we haven't BXT recalled the velocity accord yawner lasting I did something here I can't get away with just biting things by decay but divided by he squared so I take outside for dt GAC did not do it In this can be Greek in spacetime another space time we Greek indices Evelyn but because you only have each 1 of these components it is equal to 1 man there is a or it sequel toward velocity along the trajectory of awards a thing like XTT along the trajectory is a component of the velocity of the particle all are just 1 so each 1 of these things has a definite value along the trajectory a certain function of the velocity of function of the velocity and the position where a square of things function of velocity and position kinds DT this now has the meaning of the conventional integral along with trajectory here early in the grand who and the trend is below ground here it is a thing which plays the role of a grand jury and action by definition is equal in the role of the law granting which is a function of velocity and position pretzel you treat we returned to a problem that we saw all quarter was last quarter to quarter who quarters ago the problem of classical mechanics how to calculate the equation motion from 9 from action that I'm not going to do it here Lebanon tell you that if you work out oil Lagrange equations me remind you what they are transient a definition now minus and the square root of minors Genia know of X X new by town PX nobody coincidental uses is the thing inside the square root is the thing inside the square root positive so the didn't think a square root covering signed here as positive and positive this it is the proper time and it's always positive for a time like trajectory particles always move on Mike Gehrke real particles do not move fast speed of light and circling case unless you missed a space like trajectory is 1 which is going faster than the speed of light so this is always positive inside here for a time like trajectory is grungy and I want to do it now then I received a Bird said just because we had now but but for a definition the modern definition of our June asked the minimize the action is equivalent to applying or the Lagrange equations granted what's right and that remind you what they are the derivative of the Lagrangian with respect through our court X new dock at the time the next new all incidentally on again be keyed is just 1 right so deeply by me just ignore that this is a function of the space component of the velocity and X itself right do we take brandy and we differentiate with respect to a of velocity and we think it's time derivative and said that people could be L by DXM this is 1 Lagrange equations motion that we learned all about in classical mechanics at the point is if you know the that correct you can work out an equation of motion for a particle was motion will be a G the definition of a geodesic in minimizes it minimizes the you have the proper time between 2 . 9 others every late this definition over here well if you work out exactly this equation would be given that trickier you will discover exactly exactly these equations here so they will exercise work out the or the Lagrange equations and show that they are the same as Our areas requiring be trajectory to satisfy this equation here is is generally true that this is a much easier thing to work with the wouldn't do it tonight workout equations of motion for particles in a particular matter that will work it out and see how it
works method will be everybody's favorite we watch watch market rats were now from the problem of studying the motion of a particle or studying electric emotional particle in a real gravitational field the gravitational field son of a black hole In other words a massive object massive spherically symmetric object no outside the mass of the objects were far away from the rest of the objects were interested in the metric method of space OK so prefers I'm write a formula and they will check and see that this formula really metric really does make sense In this sense that would give rise using these equations of motion it would give rise to something that looks very very familiar namely Newton's equation for a particle moving in a gravitational field at least 20 of far away from the gravitating object Libya where the gravitational fairly weak Procter he is going to be the magical right down the let's say what's 1st start with flat space-time as just have geeky squared minus 1 overseas squared the X squared and now the expletive stands for the expo opposed to war Square D's you square so I won't write them all back now that would be flat space and we expect that as we go very very far From the gravitating object that spacetime looks flat we go forward but now we know that we could gravitating object does something space time so this should be exactly right let's put in here 1 that will be there when Yuri far away and now works put in something else and will call plus plus pass from my Mike Ms. when the word plus yacht went bust twice you will of position you was a function of position divided by sea square times DP squared everything else along the moment won't change anything else 1 post to you out they will you was right now the gravitational potential due to this object let's check that as long as our particles are moving slowly that the geodesic equation here Grosjean equation motion just becomes Newton's equation for a particle moving in a gravitational potential you will that lets let's see what we get for Haviland Jim now we can write relatives the miners and current square root OK but go back this form over here my guests 1 post to you will oversee Square D squared that I think plus 1 overseas Square D X squared plus Y squared busy for the but let's just Quebec squared the X squared not sorry beyond the action the action now the action is this thing right square Virginia PX mood by but will be for said they wrote down what's here Jimmy new DX no that's warned us to you oversee squared he squared Barbara square swarm squared the BX Quebec he squared if you don't understand where coming out square by decays Quebecers want so that's trivial the MDX but deeply squared with just call X stock squared their 3 of them early and we have it expects quick was 1 the a Dutch court in there that's our action that now NextStep let's work out now before we work it out what's approximated that's approximate we could work this out as India's lawyer I'm interested in it is very slow motion I'm interested in very slow motion because I'm interested in slow motion images in slow motion because someone so that this really does give rise and equation in the nonrelativistic limit the matter relativistic limit is 1 in which seat is taken to be very large now I could just erase anything with a 1 oversee square a would kill leader would tell everything right artillery at the a something are here Martinez M C squared from really putting sees in the action has an MC Square and embassies greatest as dimensions multiply as it is our action now that expanded for small quantities what is it Ressam John getting confused about the bad news is really mind this year How on note plus this might this but critics OK how do you what you do with a thing like if you are
approximated wants to use large well you right thing inside the square root as 1 class something smaller something small has a white honesty squared and and then what is it gets to you Midas next stop square brings MC Square many of the binomial expansion you the binomial expansion the square root of 1 plus a small quantity is 1 plus a half a small wanted racers becomes minus MC Square crimes 1 now class 1 0 with to see square kind told you minus X . square finally we look at Paris has a constant minus MC Square at constant in all of our energy and makes no difference so the only thing you do with a Lagrangian differentiated so you differentiate this term makes the difference the rest of it here has termed my nest C square cancels get and took cancels Elliott minors and currency EU the other half of it has Lucy squared provided by Tutsi Tutsi squared plus am over to X . Rossello Granzien if you expand that out small quantities in the number of mystical Muslim that is constant which is irrelevant it has a car rental kinetic energy warm have them square and has a pencil and as a potential energy this is just a function of X as a potential energy which we get to chew we chose whether we like the potential energy there incidentally from gravitational problem the potential energy of a particles always proportional man reviews the potential energy proportional merits if you work out From this if you work out from the equations of motion of costs would you going to get his just no equations for a particle in a force of a gravitational field of gravitational field you what equations be there will be extra will not AMEX about is equal to minus M D you by DX that's for us but here's the acceleration of the mayor's canceled OK so that main point here was that this action which is equivalent to the geodesic here is easily work that would just by thinking of the or Lagrangian equations and it seems easier to work out if you're in the nonrelativistic limit we just say is very very large policy squares small just expands Frankfurt an ICC easy Pincus but the important point is that we we learned that there is some 1st approximation small quantities Jean or not minus gene or not is approximately perhaps exactly equal but 1 was twice the the gravitational potential the vibrancy square sure he is always learned we learned the the metric metric now cost and maybe additional playing in here which are even smaller than 1 oversee square this example oversee fall OTTAWA oversee sex or anything like that and you can see it would not be important in the nonrelativistic limit so we can say with complete confidence the G nor nor was 1 plus twice the potential energy particle abide by sea square but we can't say that must be be true tool the 1st order in small quantities small quantities means war overseas work but so attractive and now works right down the middle of the room in your ride again but after wanna do is you will put the mass of the sun summers of the sent there the mess of the sun provided by the distance of the sound in front describes whether might decide this the standard gravitational potential energy the massively gravitating object G divided by distance away your Rex and now I can write down offers guests at the Met trick of space time surrounding the gravitational met areas he cows squared leading order is equal to 1 Martinez true with Emma G aura C squared the from here at the the and of RC square house could squared minus DX glare mind as to why square minds busy square that's cow court and of course plus overseas quarters plus smaller thing further beginning the familiar 29 recognizes peace watch which were chilled medical metric of a black hole but not quite C squared horror here Mary next let's take DX square post BY square was busy square as the is that just ordinary metric of three-dimensional space and this formula over here only the time component of metric In fiddled with the space-based components of the metric and everything else not yet been filled with they will pay but not much IPX let us BY Squibb was busy square is just the usual method of about ordinary flat space let's take flats space polar coordinates coordinates three-dimensional Pola Court three-dimensional polar coordinates characterized by a distant namely a distance from the center of the case dissidents and a pair of angles the pair of angles Kabila polar angle an azimuthal angle so states works drove fear him point on the surface of the sphere the sphere Is that distance all Our is
in angle an angle From the from the plane from the from the horizontal plane here we think to be the angle from the North Pole but I forgot figured to be the angle from the equator the horizontal playing here the equator angle from the equator is called they don't and enthusiasm youthful angle our there can be measure from have the fight fine OK now what is the link element on the surface of a sphere alike on the surface of the sphere of radius the lenders or Abedi want houses magic expressed in public or right for you 1st of all has br squared incidentally of the took them mentioned you would know what the write to write br squared class Our Square D favor square but what you're right through would be br square foresaw squared status quo and that's their that's correct but is another and the cosigned of data Square D 5 square this is the magic of for nearly 3 dimensional space in quarter go angles and a radio and radius era square square YLE don't polar coordinates because Paul coordinates are the good coordinates for studying the central force problem although coordinates a convenient you don't his ex-wife disease a study the motion of a particle in a gravitational field you want to use the the polar coordinates soda this metric in terms of polar coordinates have it now those make 1 change would only give the status quo Air Force cosigned favor square defy Square which is gonna given names of right over and over her Colin D. Omega square for another reason and cannot have the right over price arena however the magic of ordinary space are squared the Omega scored and that's what goes here exactly what goes here so what's now we write question the question of what to rewrite there's 1 overseas where they are squared this minus 1 overseas square are squared Be Omega square and this kingdom mind foresters revealed is nothing special this complex thing here is just a metric on the surface of the if you keep are fixed and you move around in a tangle this gives you the contribution of metric from moving in angle Bell will while notice something terribly wrong with dismay that some very very wrong with it it's find when you far away but something deeply wrong when you get close to the center what happens when all are becomes too small when becomes too small knots Zero are positive but too but the more when on becomes more this becomes large 1 minus the large number of becomes negative this coefficient becomes negative these coefficients are also negative what happens to the Met when you move into close where has all or positive eigenvalue or threat of force got for positive ID values instead of 1 negative Eigen value and 3 possible moving took this quantity changes sign with the change signed it changes signed when are is equal to what MG overseas square the factors a nice away for right threatened the way let's write it and army are going to MG overseas squared provided by far remember bracket but Oral Laura's 1 2 in overseas where the by Byars's not us what happens when aren't gets smaller than to Angie overseas with Wandie overseas with just a number of positive number that some particular characteristic radius of the black hole Mrs. watcher at some point where are moves toward the center this change stock becomes negative or happens the sign here becomes exactly the same as the sign here will make it the meaning of that as we somehow heads in Pool region where there are forced based directions and no time direction that's bad that's bad that's something we don't want good thing what really happens you study the field equations and General Re which will not do what do I mean we just tell you what the right answer is the right answers it is at a point where this change is starting this 1 also nothing here yet I have given you really put the world see square Bolganup put something else he said has the property that when this want changes starring this 1 also changes sign so what happens this 1 turns into a space actions but this 1 turns what kind directive b signature of the metric is maintained 1 time like direction and 3 space like their so what can we put here the main this flip side in the same way that this sign will you could put in 1 minus and G overseas where foreign firms are squared so and this will change signed this work that was not White To be the right thing no but Einstein's field equations you could do Frank thanks to a greater give us slightly different parts it's a substantially different dancers to work says it's 1 of minors Phil energy overseas square are I he squared appeared minus now and 1 divided by 1 minus 2 MG overseas squared horror squared minus 1 oversee square ah square Iomega square right so when we got to the
point as you move even closer the this watch O'Grady this change signed but so that's because they both change sign the signature of the metric stays the same but something very orders have been the Bangalore calling tedium becomes a space like direction and the thing calling all are becomes a prime art direction they fled to inside that accord are becomes a kind and the cordon if he becomes a space character this is not easy to visualize but I will give you the tools the visualize it as we go along something has happened as we passed in through a certain Surface Surface being are equals to and g of C squared when you pass inside there there's a flat between time radius radius becomes kind of becomes a kind variable time becomes a space this is completely mysterious for the moment but I assure you we will we will explained some geometric detail what Top your yes posed by Syrian policy I got right now near here Rujul remove burn that problem we're going to change that cross that line it takes a infinite amount of coordinate turning by a finite amount of proper so somebody falling with a particle Clark and then hand will say it takes in a finite amount of time to cross that line somebody watching from the outside watching that person for rules say it takes an infinite amount that's a characteristic of the method in the for of things more work out tonight was later changed both yesterday it is not based on changing the coefficient is to make sure that the metric space with a signature crossed my my on my passport if this when changes sign something has to give were the other ones in the natural candidate is we are here that flips signs are not think about where this is happening this is happening at a distance or coordinate are to Angie overseas see is a huge number he is a huge number and the number may miss it is a small biscuits which is a small dose but a small distance but that depends on the Mets small distance depends on raft could burned MySpace and would be about centimeter soaked with none of the worry about thinking about the gravitational theory is that it was a game us Is this a not a not sticking sticking in form clocks clerks don't care about what coordinates you use we're going to discover that this is an artifact of particular coordinates nothing going on special at this point we discover that this is just awkward coordinates them make it look like something's funny is happening here switches basin nothing funny happening there and work out see exactly why but for the moment it does look like something going on square are does look like something going on when this thing vanishes when this thing managers yet this thing becomes infinite looks like some terrible things happen geometry and factor geometries completely smooth over nothing special happening on the light cone is perfectly healthy on all in all regions and hear nothing funny going on but we have to work harder to see this was the former that's what you derive which watch over which would you prepare Einstein wrote some field equations Schwartz sealed and what Einstein knew about them as he basically knew this piece here and he knows some all with the does 1 thing the support you want your which I neglected say how big is this correction here well if I want estimate how big this correction in nonrelativistic limit I want seem to be very very large if is very large that I should expand this this 1 over a small quantities this is a very small quantity toe Angie oversee squared off services approximately equal the 1 plus 2 MG policy Square aura past Corrections date 101 miners X 101 miners acts is Eagle the 1 plus plus X squared plus 6 killed up up up up up a binomial expansion and so the 1st correction arrived 101 runs 2 and you are is 1 plus the small quantity the next contribution is to MGA over a C squared off squared this is typically small cannot they cannot closer to the center of things if you're a reasonable distance away so the OMG Morrissey square is very small the next terms of smaller and smaller and smaller notice Arroyo Wabbaseka bought receipt of the 6th Wabbaseka a so would in here is where modifying this he also With corrections which are very small the Mexico ELECTION he does contribute to the motion of particles and so forth but not in the nonrelativistic limit next correction here would be something of water while receipt for the X squared Loblaw Bob Seiko for so as not don't get To Close well hours small this correction is pretty negligible a Torah Wabbaseka before should be BSE square here should should be a C squared yet even this Turner even this term small even this time a small because they see squares this multiplied by another C square so even a small too small to be imported I don't think I convinced you that I could throw the term away Her parents do it again this it is 1 of of 2 MG Rover C squared off our next turn is to end 4 and squared G squared off square 1 of receipt the got old that's multiplied by 1 oversee square all its multiplied by 1 of the 2 square and noticed that even the 1st correction here is 1 overseas for the fall off much smaller than the original turned BP squares even these things a small month especially the correction was small thing is even more so this is negligible compared this knowing we simply don't have to worry about the slow particles particles moving much slower than the speed of light OK but that's let's a combat record here before we analyze it and see what's going on at the point where you cross from space harm will you cross the horizon of the black hole let's use it let's use it and see if we could figure out how particles more none of the nonrelativistic limit but that's the better but see if we can answer how particles follow without taking any approximation of Ramat money approximations incident we reminded let's see people who warned let's sexy Cooper warned keep in mind that we set steeple the warned said Siegel the wants just a choice of units and now Stevens see how particles moved in the presence of the Mecca Asansol hard reviewed consider Witting consider took kinds of who kind of orbits of our work out for euros or show you have word circular orbits will find a bit of a surprise about circular orbits and then radio or radial orbits many of the review orbit before the review of the 1st round of the circular orbit which would you rather see circular orbit would lead the bulk of the book 1 flowing radially and a son Robert is son or number or just a crash
crashed fractures are here's our on object they rocket crashes into river and the question is the question was ready asked how long does it take a rocket to reach the horizon along our personally arise now but but now we get a lot of talk now about the extent of the external observer who uses teacher aim whereas somebody following in the rocket uses house How time proper time in the rocket work were current from the outside summary watching their clocked somebody and outside watching this thing with their clock uses time t rental somebody falling in uses profit for Pixar weaker is both questions and get different very different answers OK both of the main point of this is to show you how things work so let's go back for Lagrangian in falling apart from particle where is it uh let's go back to the place where you start with any such calculation like you start with a Lagrangian particle and you work out for the Lagrange equations or whatever you do with and there going you solve the equation they're attracts trip to canoe tricks that allow you to do problems OK program Yap reaction is minus and relieve the sea square that restricts eagled the warm Kansas square root of deep square Mosquero Depass forever Pasqua over here is equal to nearly right here because cause I'm also going to assume that the particle does not change its anglo when it comes in it comes in all along a particular angle and we will not have to worry about the peace of this over here we're not moving around in angle so we're not moving around an angle as the particle flows in reconsidered towel is equal to square root of 1 minus 2 NGO Va he squared minus 1 over 1 minus 2 MG all are they are squared but put this into here this is the action in this into Rhodia Burnett to make sense out I'm going to divide by decay squared multiply bt team 1 murders analysis quantity they are by deeply square as just the velocity the radial velocity square are dark square so this is are about squared here and I have a Burns is my around June L is eager for minors and square root of 1 mind 2 NGO minus 1 over 1 minus 2 I'm the are God squares the would live in this cases are here we have branches OK fortunately this problem is easy or lease it's easy to see what happened with some worry about the exact solutions exactly solving it is a little bit harder but seeing what happens is easy noise a conserved quantity what so what's what's the court wants a conserved quantities always there energy right pledges what is energy in terms of Grand Duke Energy in terms of land use the Hamel come so we remind you what the Hamiltonian Is the Hamiltonian the 1st thing we do is we calculate the momentum conjugate are works the momentum conjure guitar parcel of it L with respect are . remember that are still available with respect respect to the velocity is the momentum conjured are and if we work that our approach you to what we get foolhardy and the square root of annoyance earlier annoyance but not a lot of terrible don't think I will work out I'll let you work past with respect our diet is a straight forward thing you just differentiate with respect the and you get a piece of ah what he'd do it it you multiply it by Aug . and then you subtract the Lagrangian parcel with respect are you multiply by are about to subtract the Lagrangian and that's called the Hamiltonian where the energy that's the energy of the system and Yukon working out you work out remarkable work areas each is equal to 1 9 2 0 are crimes the Mets the violated by square root of 1 my to MG are in brackets current are about where they received minus no 1 1 might use to MG are duck squared complicated thing it's an ugly thing it's a complicated thing but definite and what is it depend on it depends on the velocity depends on the velocity and that's the energy what O'Neal about the energy we know let's consider doughnuts conserved and so we can write but this is equal to and that it doesn't change what kind does not change with time and let's see what that says that tells us that we can solve are not confined their work are about used as a function of the energy and work are about use as a function if urging just follow are not solve this equation is that it is much easier than it looks you just Square Square era the square root of East square the left-hand side of the square this year and turned over and calculate are are here visitor gets its rather complicated part about it is approximately equal to 30 works there are going to Judy over it's easy part of its exact or it equal to 0 breezy airport but squared a energy square firms 1 minus 2 NGO squared minus 1 minus 2 and G R Q provided by the square of the energy that doesn't really matter what it's that's a good thing but what does matter it is what it does when authorities in year to energy when is to MGU something bad happened or hard Doc is
approximately equal square root of all our tool injury provided by 2 Angie energy is energy is a radius of the horizon Oracle's to Angie what this is telling us is article falls below the horizon is velocity goes 0 0 has a gift through force was the Schweiger radius and velocity smaller and smaller instead of accelerating it gets smaller and smaller and smaller see that from here or there are I the yup the velocity gets smaller and smaller it had no dropped knocked on his knee over others fast so you here Pomeroy this goes 0 goes 0 observer of the to Tour Obama this goes 0 0 0 this will 0 the 1st hand here also rose 0 and mohair lived the marmalade a hard about squared all times were won over this thing here square so this goes to infinity man is 1st day of the week now does she had a view of their thus we would reveal Yasser this goes to infinity right so if OF aren't that did not go to 0 this would be 0 over Infinity which will be 0 but can't be 0 it's supposed to be the energy which is conserved energy is conserved and 0 I'm not cannot start the only way to avoid the makers B 0 0 make sense that would be that are about goes but this is you could solve the equation soccer staff are about as a function of E & am and everything else and this is what you get so was boasted to are doctors that's a surprise a surprise you might think of thing accelerated but instead it slows down slow down slow now we see why this is slows asymptotically slower and slower and slower and never passes the point repairs to the point of asymptotically approaching it takes forever to get here Causey why this series I wonder think we can on Let's go back to go back which were children OK the Swatch electric is power squared is equal to 1 minus 2 NGO are key squared minus 1 over 1 minus 2 injury of br square let's forget the other pieces the of the it is not playing any role now how's a radial light rays from radio library which is moving in for the year or a buddy movie always moving always think about like moving away from the object at distance are healed moving away of flowing doesn't matter Boeing in or moving away look at what is a light ray of light rays as a solution to the equations that the the interval here the distance here is like life like like means D House where people 0 are original original like like means deep house where it is equal to 0 0 some white Ray satisfies but fathers this equal to 0 and that will tell us that deal that would they are busy with times 1 minus 2 MGU Hobart work to do it this eagled a 0 marriage multiply both sides by 1 minus and give us 1 minus 2 energy of squared deeply squared equals br squared the square root now provide by PT that's a light rain as the fastest thing they can ever be a wiring nothing can be fast the library falling in this to be falling in Iran might be the positive and negative and in library normally find that an aura goes the wristwatch O'Grady happens the fish watcher radius fiscal 0 this is equal Carmine's to NGO our and that's equal the radial velocity R dot the magnitude of the radio velocity even a light gray beard stock as moving toward visible only with the speed of light but also the moving with the speed of light was but the speed of light has this property that in these coordinates In this particular set of coordinates the speed of light goes 0 as you get closer and closer so nothing even including a light gray can get past this the surface here was solid or so it seems that I would be much this is a much easier way to think about but to stay or go down now what about went far away have known as far away as far away as 1 the most important thing 1 far away they are as 1 1 here means a speed of light as it starts moving even closer and closer and eventually gets to the point where this goes to 0 and then the RDP simply go through your something odd happens beyond that point something odd happened on that point and work that out OK this was I'd at least the problem with general views principles a pretty simple and the complications are always on they almost always are going there there is is nothing nice about the actual computation principal simple and they give rise the complicated equation where the US poor with such questions now price there's with that also so they reached text is said but obviously forward right at that would get anywhere it so as well what is the use them newer Member escape from the back yard young distracted Our This is the magnitude of velocity magnitude velocity goes 0 b the going or outgoing radiation our let's see what we do here we had we had an equation that this is equal to the square before I took the a square root now there solutions when I picked square root the solutions take the square root of plus or minus a plot solutions is out Gordon radiation outgoing radiation going out going like moving out the radio velocities positive negative solution here light
falling in but in either case the magnitude of the velocity goes to 0 so you're right if you start a light rain right at media the shorter reviews it won't get anywhere in fact a library will just sit at the rubbish what a light ready interior of the short-term rates will fall in a library exterior will go out but the closer it starts the young British watch radius the longer it will take to escape purposes the materializes outside the black hole called the materialized but as they will but outside but if it starts out very Klaus who will sit there for Mary very long time and then after a very long time and it will be observed to be a little bit away in more away more ways than call for you right they were was the 1 where the poker it as they were about not viewers the rumor wives warning a rumor Of course they say they got entries is it's apparently stationary weather find out that that that from the point of view of somebody falling with the photo on the time of the speed of light and everybody including the photon disco sailing right throw Rice's an artifact of peculiar court Verizon nothing really going on with a stranger proprietor torture rape 1 other thing that I should emphasize is that we really talking about the sun or the earth the earth well the earth is how big where is 12 thousand area kilometers a success and kilometers an hour and radius around 6 thousand kilometer radius Miss watch over reviews for the Earth is about 7 but the thing like that and this means that you can you don't trust the solution fury with real stuff this solution His only correct outside the region where the stuff with a matter which creates a gravitational field so this is the solution of the equations of motion for a lump of material outside a lot now only of the lump it shrinks and becomes smaller than the previous year that this radius means anything particular only if the whole thing work collapsed radius smaller than the Schweiger radius of the short story that we know that the solution is meaningful and correct emphasis small distances saw nothing is nothing funny going on at the center of Europe in particular even this thing which I'm telling you is really not so funny in this this does not happen near the center of the center of Europe does not having a horizon rather the solution is different and the interior when it is on outside piece together but if the earth the collapse and shrink the small which water reduce would the would be pleasure you false my goal is going to say From last watching it in a sense that we will we will not receive crashing we will watch it asymptotically gets FLORENCE FLORENCE FLORENCE down never passion whereas somebody writing with falling material will just right fortunately gives us last server I get to see their or else outside could never fight back for what you just exist try not but the al-Saud observers said nothing versions of black hole area largely because we had a good Alcide test us ride was across it s should call could not that new for this practices would saying sound right but is but on this day the understand how did you might say that a black coal Connecticut started that's a problem that start with only a little trained black hole yes let's start with a tiny tiny black called and analysts throw stuff into it and try to make it figure as a people Alaskan happen the shiny black locally grown from more planets and blue-black will eventually will grow from being a solemn mass size the big news outside of growth but I just told you also that nothing ever gets through variety of something not quite right it would work out some examples of exactly how a black hole grows answer the 1st World at can use jets not around so what want no you radio none will tell us what is this 1st there that's yesterday who roulette but will of them's area that's right yes certainly it stopped short faster and faster Phillips starts to grow slower and slower point of point a gun was point is a specific point where it's and overstuffed looks launched but the radar gun as he arrived now so as is falls flattens out must flat because the 2 edges of at the front of the back of it never get past the horizon the front and back get past the horizons both asymptotically moved toward it gets it gets more more squashed but now the radar gun sins in its radar and radar does the same thing and made I never captures the radar will never catch me in falling it will also slowed down so after a long period of time your car will be squashed against the horizon here and the radar
beam will also be squashed against both be inching slower and slower and slower for the horizon with a radar beam never quite catching the horizon yes yes yes just to the Glomar ruled a quorum the radar being bounces off it will be enormously rich the but not heard exactly not knowing what I said about the radar beam that catching up would be true if this were flowing you if this were worldwide but yeah just takes an enormously long time for the radar beam catch job when the rate being reflect the enormously rich then we met raiders Cromer 4 more days he or she is which is where young up for sale I that's what I wrote that all ensure or what was said now what is true let's again we have a somebody falling use the car and car squashed Maryland them the so we could ask the light gray which is crossing cars car falls in it gets more and more squash and at the same time times slows down we haven't talked about the slowing down of perimeter here kind slows down meeting because steady warned this quarter a few days away think about this quantity is going 0 when this quantity goes 0 giving amount of DT corresponds to smaller and smaller and smaller proper in the car so will will get to to it will work goalless at work but what does happen is the light gray when you take into account how the clock behave and how the meter sticks behave and everything the person in the car will see light gold from the back of a car from the car with the speed of light time clocks meterstick Everything will expand contract whatever they have to do and in court following with the car light will go from the back of the front in exactly the amount of time girl that's that's not going to change Carter go Will worker who work about work your way the chief grab a state based on how would it be wrapped up a on is its fair share of the other forces that are theory that states most ideas about unifying gravity together with the forces involves getting other dimensions space when you them into the space the gravitational field becomes more complicated because more components and the additional components of the gravitational field view from 3 dimensions 3 3 space one-time dimensions of the components become the fields associated with electromagnetism associated with the standard model himself or so from the point of view are 3 plus 1 ordinary 3 plus 1 dimension because one-dimensional three-dimensional base when the time you have a bunch of fields of editions of the gravitational field when they put together higher dimensional space all you have is the gravitational field so that most attempted unification work along warned that additional direct space motion the different Erickson the additional directions spaced our give rise properties particles for example which other charges which various properties and that's where most the ideas that unification Sony and toll gravity but gravity and get us series there an endeavor resulted in 1st it's the the signature forward but others just In here with you Conestee whether the current versions of the year you ask whether reality reality and nobody has the vaguest idea could be regions a space with Bob properties of different signatures I have no idea why are so that some slate the signatures sitting there co-ops which it's more than that it's a classical classical property of general secretary general of the equations of motion will resist the change in signature as the signatures tries to get to a point where it's corporate change the energies becomes an explosive it's kind of like trying to exceed the speed of light energy blows up at the point where the like approach energy blows up there the signature of space times it close to the point where it could ship sold it's more then that's more than the statement of empirical fact that a property of Einstein's general theory on the other hand you could ask Well what happens if you'd end to Einstein's general relativity quantum corrections all sorts of things and nobody knows best nothing anybody has ever seen in any version of quantum gravity string theory anything else has ever led to a change in the signature does that mean that doesn't happen somewhere and it the way way beyond it would be severe right now her From a humanistic interests and credits were still Weiner's what were was should get inside these given that so far out so that back don't for your guess is as good as were in use of it was said to be an astronomer but he was obviously a pretty sophisticated mathematician our if you are a few hours in World War One I don't know if it boggles the mind sometimes when I was going to withdraw all of them 1 where you're going through all of for more
please visit us at Stanford . EBU
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Metadaten

Formale Metadaten

Titel General Relativity | Lecture 5
Serientitel Lecture Collection | General Relativity
Teil 5
Anzahl der Teile 10
Autor Susskind, Leonard
Lizenz CC-Namensnennung 3.0 Deutschland:
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DOI 10.5446/15035
Herausgeber Stanford University
Erscheinungsjahr 2012
Sprache Englisch

Inhaltliche Metadaten

Fachgebiet Physik
Abstract October 22, 2012 - Leonard Susskind derives the spacetime metric for a gravitational field, and introduces the relativistic mathematics that describe a black hole. 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 and in this lecture Leonard Susskind moves the course into discussions of gravity and basic gravitational fields.

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