General Relativity Lecture 5

Stanford University okay all right let's talk about time like space like and light like for that we just go back to special relativity let's uh spell out what that means and special relativity there's a metric we discuss this metric many times we call it proper time the tau squared the tau squared is the square of the proper time of some little interval and as you remember it's the T squared minus the x square minus the Y squared minus DZ squared and if we want to put the speed of light in sometimes I'll put the speed of light in the reason I put the speed of light in sometimes is to keep track of what's small and what's big under certain circumstances for example if you want to study if you want to go to the nonrelativistic limit the limit where everything is moving slowly then it's good to put back the speed of light because it keeps track in that case of the fact that the speed of light is much bigger than any other velocities of the problem so let me put it in for the moment I'll put it in and take it out right there on the various circumstances it goes in like that 1 over c-squared and if the tau squared is greater than 0 if the tau squared is greater than zero now of course the square of a real number is always greater than zero but the Tau squared is not defined by being the square root of real number it's defined by this combination over here and it can be positive or negative depending on whether the x squared plus dy squared plus DZ squared is bigger than DT DT squared or smaller than DT squared I said the x square plus dy squared plus DZ squared but I meant the x squared over C squared plus dy squared over C squared and so forth but in any case if the town squared is greater than zero then the little element here is said to be time like it's got more time than it has space so to speak this part is bigger than this part and that can be described in terms of a light cone remember the light cone the light cone looks like that and the boundaries of the light cone the cone itself are the places with the T squared let's for a moment I don't want to see squares that DT squared equals DX squared plus dy squared plus DZ squared that's called light like if DT squared is bigger than the x squared plus dy squared plus DZ squared it means the little vector lies in the interior of the column like that it could also be lying in the backward direction in either case a little interval or a vector let's speak about a vector in general a vector in general is time like if it's time component is bigger than the sums of the squares of the space components okay that's time like space like is exactly the opposite its DT squared the tau squared less than zero and in that case we usually define another quantity the S squared which is just a negative to the tau squared and it by definition is the X square plus dy squared plus DZ squared minus DT squared the same object except for the relative or the different with a minus sign space like vectors are ones for which the S squared is bigger than zero the tau squared is less than zero and space like vectors lie outside the light cone finally there are light like vectors it's a space like and finally there are light like vectors and light like vectors are the ones for which DT squared equals the x squared plus dy squared plus DZ squared those are the trajectories of light rays and they lie in the cone they lie in the surface of the cone itself those are the three kinds of vectors that you can have just for a moment consider what it would mean if there were two negatives our sorry two positive signs and two negative signs instead of one positive sign and three negative signs somehow this would correspond the two time directions there that doesn't mean anything in physics and physics there are never two time directions there are always one time and three space now can you imagine a world with two pi can't frankly I can't I can't imagine what it mean to have two different time directions and we will simply take the view that that that's not an option there is always one time like direction in the metric here and three space like but that doesn't mean that there's a unique direction which is time like there are many time like directions pointing within this light cone here the invariant statement is a statement about the metric tensor the metric tensor we can write this in the form the tau squared is equal to G mu nu with a minus sign DX mu DX no the minus sign is a convention it's always defined so that G mu nu the X mu DX nu goes with the space like vectors having a positive sign here incidentally one other thing or just remind you um the proper time along a trajectory like this is a measure of what a clock along the trajectory ticks off the proper time is a kind of time it's the time read off by a moving clock proper distance the s squared proper distance along a space-like surface is distance measured by a meter stick so proper distance the S squared or the square root of the s squared really is a distance and proper time really is a time and keep that in mind okay so this is our definition of the metric and for the case here that we've written the metric can be written in terms of a matrix and let's say the matrix is minus 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 it's called 8 mu nu it's like the chronicle delta except it has a minus sign okay this thing has obviously three positive eigen values it's a matrix it has three positive eigen values and one negative 4i ghen value and that's the invariant story that's the story not that there's only one time direction but that there is one negative eigenvalue here and a metric which would have two negative eigen values well 3 negative eigen values will have more than one time and we just don't even think about that that something that's physics does not seem to have made use of to negative x ok so that's that's space like time like light like and so forth more generally it could be in flat space but certainly in curved space curved space-time the metric tensor G the metric tensor cannot be chosen to be equal to this ADA symbol here it's a general function G mu nu of X or g miyajima mu of x cannot just be any old thing at every point that every X there's a a matrix at every X is a matrix every point and every X the there must be one negative eigen value and three positive eigen values wherever you are standing there you should experience a world with one time direction and three space directions or more exactly one negative eigen value and three positive okay so what that means what that means is that every point in space has a light cone associated with it now I'm drawing it sort of tipped and tilted that reflects the fact that the coordinates themselves might be wavy and curved and so forth so as you move around because the coordinates and also the metric tensor might be varying from place to place the light cone might have an apparent tendency to look different in different places but that is just because some because as you move around in space the metric tensor may change from place to place and the coordinates change from place to place all right so at every point there's a light cone there's an ocean of light a space of time light excuse me I'm like vectors inside the light cone space like vectors outside the light cone and like like the motion of light rays along the edge or the cone itself the cone is a surface when you have a ice cream cone the ice cream and the inside is not part of the cone the cone is the surface itself the shell okay so we have to make sure then when we write a metric that it does have this property of the right number of positive and negative eigenvalues incidentally it's called the property of having a certain number of eigenvalues positive and a certain number negative is called the signature of the metric one says the signature of the metric what is the signature of the metric of ordinary flat space the blackboard not space-time the blackboard has two dimensions and the signature of the blackboard would be plus plus the x squared plus dy squared the signature of Minkowski space special relativity is minus plus plus minus plus plus minus plus plus plus excuse me minus plus plus plus okay so we have to make sure that if somebody if somebody gives us a metric wherever we got it from we might get it as a present in the mail we might we might calculated from some equations of motion some field equations always we should make sure that that metric has signature minus plus plus plus and if it doesn't means we did something wrong not only should it have that signature every point and at some point it should have it at every point yeah there is a single eigen eigen vector associated with negative eigen value but no um no no no this um there no there is none oh there's not a single vector there's not a single eigen vector anything anything in this interior of this light column is equivalent to everything else by Lorentz transformation you can Lorentz transform this vector to any other vector no this nano you see that that's incorrect supposing I gave you the metric plus plus plus plus does that look like it has only one positive eigenvector sorry does that look like it has three positive eigenvectors and everything else is not the nagging vector no any vector if this is the unit matrix any vector is an eigenvector vector with plus one here it's true that you'll work this out any time like vector is an eigenvector of this matrix with eigenvalue -1 any space like vector is an eigenvector of this metric with might with plus 1 in any light like vector is an eigenvector with 0 ok you can check that that's something to check well alright let's let's take this let's take that good let's take this light cone over here this light cone with XYZ t coordinates defined in this way the light cone is a 45-degree cone with the with the generators of the cone the lines in the cone here are 45 degrees if I were to multiply let's say DT squared by some number what so now let's let's let's divide this here now it's the same space it's exactly the same space it just corresponds to changing the x coordinate from X to X divided by C ok all right but if I now drew the light cone in the coordinates X and T it would look like this if C was large will be squashed that's just reflecting the fact that light moves of the speed of light and so if C is large light moves a long way in those coordinates same space have not changed the space just change the coordinates but by changing the chord so I can squash the light-cone i can even tilt the light-cone i can do all sorts of things to those core to those like cones it does not reflect a real change in the geometry all right now since we have just arbitrary coordinates that we're dealing with here we could for example have this speed of light here depend on position if it depended on position it wouldn't change we could have it depend on position then the metric would have some funny position dependence and the metric was you moved from one place to another we go from looking this way to looking that way so it's a pure coordinate issue whether it looks one way or the other but the invariant statement is that there are space like vectors timelike vectors light like vectors and the boundary between the space like vectors and light like vectors is a comb space like a vector is a 3d space but time would be just one no the key point here is that that eigenvector is not a tensor it transforms under Lorentz transformation it transforms under Lorentz transformation let's see yeah yeah yeah I didn't want to get into it what you made you're right you're right but let's let's let's come back to it let's come back to it I want to get I want to get through to the metric of a black hole today and I didn't want to spend it all on this ah let's come back to it one eigenvector three one positive eigenvalue sorry one negative eigenvalue three positive eigenvalues this metric transforms under Lorentz transformations into itself it transforms under Lorentz transformations into itself so if I do a Lorentz transformation we come back to exactly the same metric and that's the important issue here or I told you what the rule for geodesic was last time or time before it's that the covariant derivative of the tangent vector is zero as you move along it moves parallel to itself I'm going to give you a different definition tonight which in many ways is more useful but let me first write down the original definition the second X mu by D tau squared that's the Durrett the tangent vector is the X mu D tau this is the derivative of the tangent vector along some curve and should be plus or minus I think plus mu Sigma Rho D X Sigma D tau the X Rho by D tau that's the definition of a of a geodesic but there's another definition which is many times more useful it's the analog of the definition of the geodesic in ordinary space ordinary space it's the shortest distance between two points or better yet stationary stationary distance between two points stationary distance between two points defines a geodesic and so another way of writing this is to say we extrema lies or we make minimum let's take the case of ordinary space ordinary blackboard or a curved version of the blackboard we calculate the distance between two points we're interested in the geodesic connecting those two points are the geodesic connecting those two points calculate a curve take a curve and calculate its distance what is its distance its distance in ordinary space would be for each little element of it we have the s squared equals G let's say MN the X M DX n now I'm talking about space in that space time will come back the space time in a moment that's the s squared or the s is just the square root of that in other words Pythagoras theorem okay that's the that's the distance along a little piece of geodesic here and then we just add nuttier that's a curve then we add them all up and to add them all up we just do an integral that integral sign just means add them all up or add them all up let the distances go to zero add them all up and that's equal to the total distance between those two points the rule is to make that as small as possible to make it extremal we learned when we studied classical mechanics how to do that we learned only studied classical mechanics that this is a problem in the calculus of variations it's a problem analogous to minimizing the action for a particle in other words we could think of this as the action for some kind of particle moving between here and here and then extrema lies this or I'll make it make it stationary is the rule for calculating the geodesic what kind of equations do we use this Jumu'ah is a function of X here of course what is the equation called that a lot better is the equation that tells you how to minimize a thing like this it's called the Euler Lagrange equations it's Lagrange's equations for a Lagrangian which is derived from this action think of this as an action for a particle going from here to here minimizing it is like minimizing a principle of least action when you go from the principle of least action to the Euler Lagrange equations the principle of least action turns into differential equations typically it turns into F equals MA type of equations all right going from here to here is exactly that operation here's an action or thing to minimize here's the solution to it in the form of a differential equation in fact it sort of looks like acceleration is equal to something that something is kind of like a force okay so let's - the real problem we're we're not interested in distance but in proper time and proper time if I want to calculate the proper time between the two points space-time points I do exactly the same thing exactly the same thing except there's a minus sign here and that's equal to the proper time between point one and point two in space-time point one point two in space-time there it is that quantity is the thing you maximize it if you minimize it anybody know you minimize it you minimize it yeah you minimize it oh sorry America let's see um we'll have the sign right when you get the sign right yes okay now let's suppose this really corresponds to the motion of a particle that starts at the space-time point it ends at the space time point the action depends on one more quantity it depends on the mass and the actual action let's say the actual action is equal to minus the mass times the proper time between points this is definition this is definition of the mass putting the mass in there we'll find out is important for thinking about energy and so forth but the minus sign is strictly strictly definition you'll see where it comes in as we go along minus the mass times this thing here and in fact the action I think is typically yeah it's minimized it's minimized typically minimized for particle moving on a trajectory okay what do we do with this thing this thing is a completely unrecognizable object you're probably saying - so first of all not only do I not know how to calculate kind of deal with it to do anything with that or even really understand what it means what does it mean to have a square root of something times the X DX and integral kind of things we usually understand as integrals have a D something 1 D something and a function write some function and a D it could be a DT it could be at the time but normally we don't see funny things like this with square roots of the X DX all right so this is easy to to explain let's let's break up the interval into little time segments little time segments delta T okay and let's divide here each one of the or DT let's divide each one of these DX is by DT by DT some of these VX M's might not the fact be DTS VT is or T is one of the four coordinates what happens if we have a DP DT we just call it one what happens if we have a DX DP we call it the velocity right we call it just the ordinary velocity now I did something here I can't get away with just dividing things by DT but I've divided by DP squared so I take and on the outside I put DT which is your AC's be Greek hmm yeah the indices can be Greek in space time now that we're in space time we can put Greek indices in there now that were in space time and you see what we have each one of these components is equal to one if it's DT DT or it's equal to a velocity along the trajectory in other words a thing like DX DT along with trajectory these are components of the velocity of a particle or just one and so each one of these things has a definite value along the trajectory a certain function of the velocity it's a function of the velocity and the position so we have a square root of thing function of velocity and position times DT this now has the meaning of a conventional integral along this trajectory here and the integrand the integrand here is the lagrangian it is the thing which plays the role of lagrangian action by definition is equal to the integral of the lagrangian which is a function of velocities and positions all right so until we return to a problem that we saw all quarter over the last quarter two quarters ago two quarters ago the problem of classical mechanics how to calculate an equation of motion from from an action now I'm not going to do it here what I'm going to do is tell you that if you work out the Euler Lagrange equations let me remind you what they are well grandjean is by definition now minus M square root of minus G mu nu of X DX mu by D tau DT DX nu by DT Oh incidentally is this is the thing inside the square root is the thing inside the square root positive so that you can take a square root it's got a minus sign here it's positive it's positive this is the proper time and it's always positive for a time like trajectory particles always move on time like trajectories real particles do not move faster than the speed of light incidentally in case it miss you missed it a space like trajectory is one which is going faster than the speed of light okay so this is always positive inside here for a time like trajectory this is Lagrangian and what do you do with it yeah why is that item underneath the square root sign equal to the Lagrangian just because it's in the action equation yeah my definition - it's more than definition if you now ask to minimize the action it is equivalent to applying the Euler Lagrange equations to this Lagrangian so let's write them down let me remind you what they are they're the derivative of the Lagrangian with respect to R let's call it X mu dot that's the time derivative of X mu Oh incidentally um again DT by DT is just one right so DT by DT just ignore that this is a function of the space components of the velocity and X itself all right so what do we do we take the Lagrangian we differentiate it with respect to a velocity then we take its time derivative and set that equal to the L by DX M this is the Euler Lagrange equations of motion that we learned all about in classical mechanics all right so the point is if you know the metric you can work out an equation of motion for a particle whose motion will be a geodesic definition of a geodesic it minimizes the it minimizes the the proper time between two points now how does that relate to this definition over here well if you work out exactly this equation with a given metric here you will discover that it's exactly this it's exactly these equations here so there's a little exercise work out the Euler Lagrange equations and show that they are the same as as requiring the trajectory to satisfy this equation here it is generally true that this is a much easier thing to work with then we're going to do it tonight we're going to work out some equations of motion for particles in a particular metric we'll work it out and see how it works and the metric will be everybody's favorite metric the Schwarzschild metric okay let's let's now come to the problem of studying the motion of a particle or studying the metric and a motion of a particle in a real gravitational field the gravitational field of the Sun the earth or black hole or whatever you have in other words a massive object a massive spherically symmetric object and we're outside the mass of the object we're far away from the mass of the object and we're interested in the metric in the metric of space in there okay so the first I'm going to write a formula and then we're going to check and see that this formula really for metric really does make sense in the sense that it would give rise using these equations of motion it would give rise to something that looks very very familiar namely Newton's equations for a particle moving in a gravitational field at least when you're far away from the gravitating object where the where the gravitation is fairly weak all right so here's going to be the metric we're going to write down let's let's let's first start with flat space-time that's just going to have DT squared minus 1 over C squared DX squared and now the x squared stands for DX squared plus dy squared plus DZ squared so I won't write them all down now that would be flat space and we expect that as we go very very far from the gravitating object that the space-time looks flat if we go far away all right but now we know that the Kerr the gravitating object does something to space-time and so this shouldn't be exactly right let's put in here one that will be there when you're very far away and now let's put in something else and we'll call it plus I think plus plus or minus - minus minus now 1 plus y 1 plus twice you of position you as a function of position divided by C squared times the T squared and leave everything else alone for the moment don't change anything else 1 + 2 u and I'll tell you what u is right now it's the gravitational potential due to this object let's check that as long as our particles are moving slowly that the geodesic equation here the Lagrangian equation of motion just becomes Newton's equation for a particle moving in a gravitational potential U of X alright let's let's see what we get we have a lagrangian and now we can write out what it is it's minus M times the square root ok let's go back to this form over here - 1 + 2 u over C squared DT squared and then I think plus 1 over C squared the x squared plus dy squared plus DZ squared and so forth just let's just put let's just call it x squared DX squared ok that's no sorry the the action the action is the action is this thing right square root of G mu nu DX mute PI DT by DT there we are DT right I simply wrote down what's here G mu nu DX mu DX nu that's 1 plus 2 u over C squared DT squared divided by DT squared plus 1 over C squared DX square by DT squared if you don't understand what I wrote coming out okay DT squared by dt squared that's just 1 so that's trivial and the x squared by DT squared we can just call that X dot squared there are three of them and they and we have to add them X dot square plus y dot square plus Z dot squared in there that's our action now next step let's work out now before we work it out let's approximate it let's approximate it we can work this out as it is but what I'm interested in is very slow motion I'm interested in very slow motion and because I'm interested in slow motion images in slow motion because I want to show that this really does give rise to Newton's equations in the nonrelativistic limit the nonrelativistic limit is one in which C is taken to be very large now I could just erase anything with a 1 over C squared in it but boy I would kill II would kill everything then right I would kill everything I left something out here it's minus MC squared if I'm really putting C's in the action has an MC squared in it okay in MC squared is just dimensions let's multiply it by C squared R there's our action and now let's expand it for small quantities what is it it's some I'm sorry I'm having John I'm getting confused tonight this is bad is this really - here no it's plus and this is - that's correct that's correct okay how do you what do you do with a thing like this if you want to approximate it when C is large well you write the thing inside the square root as one plus something small something small has a 1 over c-squared of it and then what is it it's to you - X dot squared times minus MC squared then you use the binomial expansion use the binomial expansion square root of 1 plus a small quantity is just 1 plus 1/2 the small quantity okay so this becomes minus MC squared times 1 plus 1 over 2 C squared times 2 u minus X dot squared equals finally we look at this it has a constant - MC squared a constant in the Lagrangian makes no difference the only thing you do with the Lagrangian is you differentiate it so when you differentiate this term makes no difference the rest of it here has term minus C squared cancels we get and the 2 cancels and we get minus M times u then the other half of it has MC squared divided by 2c squared that's plus M over 2 X dot squared that's the Lagrangian if you expand it out to small quantities in the nonrelativistic limit notice what it has it has this constant which is irrelevant it has a conventional kinetic energy one-half MV squared and then it has a potential energy it has a potential energy this is just a function of X it has a potential energy which we get to choose we choose whatever we like for the potential energy there incidentally for a gravitational problem the potential energy of a particle is always proportional to its mass so here you see it the potential energy is proportional to its mass if you work out from this if you work out from this the equations of motion of course what you're going to get is just Newton's equations for a particle in a force in a gravitational field of gravitational field you okay what are the equations be they will be X double dot MX double dot is equal to minus M D u by DX that's the force is the acceleration and the mass cancels okay so the main point here was that this action which is equivalent to the geodesic here is easily worked out just by thinking of the Euler Lagrange equations and it's even easier to work out if you're in the nonrelativistic limit where you just say C is very very large 1 over C squared is very small and just explained the square root alright so that's a nice easy easy thing to see but the important point there is that what we learned out of this is that for some first approximation small quantities G naught naught minus G naught naught is approximately perhaps exactly equal to 1 plus twice the gravitational potential divided by C squared so we here's what we've learned we've learned that the metric the metric now of course there may be additional things in here which are even smaller than 1 over c-squared if for example there were 1 over C fourth or 1 over C 6 or anything like that in any of these terms here they would not be important in this nonrelativistic limit so we can't say with complete confidence that G naught naught is 1 plus twice the potential energy of a particle divided by C squared but we can say that must be true to the first order in small quantities small quantities means 1 over C squared all right so let's write that down now let's write down what we know will be rightly let me rewrite it again but what I want to do is for you we will put the mass of the Sun the Sun is at the center the mass of the Sun divided by the distance to the Sun times Newton's constant with a minus sign this is the standard gravitational potential energy the mass of the gravitating object G divided by the distance away U of X and now we can write down our first guess at the metric of space-time surrounding a gravitational mass there is DT D tau squared the leading order is equal to one minus two mg over R C squared the C squared is from here there's a 2 mg over R C squared D tau Square D DT squared - the X square minus dy squared minus DZ squared that's the tau squared and of course plus o over C squared plus smaller things except beginning to look familiar to anybody you recognize this piece some of you swatch shell metric Schwarzschild metric or the metric of a black hole but not quite let's put C squared R here now next let's take the x squared plus dy squared plus DZ squared that's this piece over here that's just the ordinary metric of three-dimensional space in this formula over here only the time component of the metric has been fiddled with the space space components of the metric and everything else have not yet been fiddled with they will a little bit but not too much all right the x squared plus dy squared plus DZ squared is just the usual metric of ordinary flat space let's take flat space in polar coordinates polar coordinates three-dimensional polar coordinates three-dimensional polar coordinates are characterized by a distance namely a distance from the centre of the Sun in this case a distance and a pair of angles the pair of angles can be a polar angle and as amutha Langille so let's say so let's draw a sphere here this is a point on the surface of the sphere the sphere is that distance R ah there is an angle an angle from the from the plane from the from the horizontal plane here we could take it P to be the angle from the North Pole but I think I'll take it to be the angle from the equator the horizontal plane here is a liquid the Equator the angle from the equator is called theta and then is the azimuthal angle R that can be measured from here that's Phi Phi okay now what is the length element on the surface of a sphere like this or on the surface of a sphere of radius R and the length the the D or better yet what how is this metric expressed in polar coordinates I'm going to write it for you and first of all has a dr squared incidentally if this were two dimensions you would know what to write you'd write the D R squared plus R squared D theta squared right what you're right in two dimensions will be the R squared plus R squared D theta squared and that's there that's correct it's still the same but there's another term and the other term is cosine of theta squared D Phi squared this is the metric of ordinary three-dimensional space in polar coordinates two angles and a radio and radius the R squared plus R squared times this why are we doing polar coordinates because polar coordinates are the good coordinates for studying the central force problem polar coordinates are convenient you don't want to use XY and z to study the motion of a particle in a gravitational field you want to use the the polar coordinates and so we're going to write this metric in terms of polar coordinates now we have it now we're going to do is make one change we're going to give this a name the theta squared plus cosine theta squared D Phi squared we're just going to give it a name so we're not to write it over and over we're going to call it D Omega squared for no other reason and cannot have to write it over and over okay so we now have the metric of ordinary space R squared D Omega squared and that's what goes here that's exactly what goes here so let's now rewrite question is a question okay let's rewrite this 1 over c-squared the R squared that's this minus 1 over C squared R squared D Omega squared and just keep in mind R is just radial distance nothing special this complex thing here is just metric on the surface of the sphere if you keep are fixed and you move around an angle this gives you the contribution to the metric from moving an angle now now there's something terribly wrong with this metric something very very wrong with it it's fine when you're far away but there's something deeply wrong when you get in close to the centre what happens when R becomes too small when R becomes too small not zero but when R is positive but - but the but small when R becomes small this becomes large one - a large number becomes negative this coefficient becomes negative these coefficients are also negative what happens to the metric when you move in too close to the center it has a four positive eigen value for positive eigen values instead of one negative eigen value and four and three positive eigen values when you move in too close this quantity changes sign where does it change sign it change a sign when R is equal to two mg over C squared in fact there's a nicer way to write this let's write it a different way let's write it as R minus two mg over C squared divided by R thing in the bracket here R over R is one two mg over C squared divided by R is this thing here notice what happens when R gets smaller than two mg over C squared two mg over C squared is just a number positive number okay it's some particular characteristic radius of the black hole it's the Schwarzschild radius at some point when R moves in toward the center this changes some becomes negative what happens is the sign here becomes exactly the same as the signs here all negative the meaning of that is we somehow passed into a region where there are four space directions and no time Direction that's bad that's bad that's something we don't want right I'm not a good thing what really happens if you study the field equations of general relativity which we'll all do we'll do this but for the time being let me just tell you what the right answer is the right answer is at the point where this change is sign this one also there's nothing here yet I haven't given you anything to put here just 1 over C squared but we're going to put something else here that has the property that when this one changes sign this one also changes sign so what happens this one turns into a space direction but this one turns into a time Direction the signature of the metric is maintained one time like direction and three space like directions so what can we put here to make this flip sign in the same way that this flip sign well you could put in one this two mg over c-squared are in front of the dr squared so in this one change sign this one change sign that turns out not quite to be the right thing that that Einsteins field equations tell you to do Einstein's field equations give us slightly different worlds it's a substantially different answer but let's see what it says it's one minus two mg of the C squared R DT squared okay - now and one divided by one minus two mg of a c squared r dr squared minus 1 over c squared R squared D Omega squared alright so when we got to the point as you move in closer than the Schwarzschild radius this changes sign but so does this because they both change sign the signature of the metric stays the same but something very odd has happened the thing that we're calling T here becomes a space like direction and the thing that we're calling R becomes a time light direction they flip once you go inside that radius the coordinate R becomes a time and the coordinate T becomes a space direction this is not easy to visualize but I will give you the tools to visualize it as we go along something has happened as we passed in through a certain surface the surface being R equals to mg of a C squared when you pass inside there there's a flip between Prime and radius radius becomes time or becomes a time variable and Pyne becomes a space variable this is completely mysterious for the moment but I assure you we will we will explain in some geometric detail what it is it's happening here yes almost I see it again almost time how much time work cross that line yeah yeah yeah yeah that's what we're going to do we're going to do that we're going to do that problem we're going to see that the cross that line it takes a infinite amount of coordinate time but a finite amount of proper time so somebody falling with the particle with a clock in their hand will say it takes in a finite amount of time to cross that line there okay somebody watching from the outside watching that person fall through will say it takes an infinite amount of time that's a characteristic of this metric and that's one of the things we want to work out tonight question yeah what was the motivation for changing the coefficient of the r-squared just motivation for changing the coefficient is to make sure that the metric stays with a signature plus minus my R minus plus plus plus okay if this one changes sign something has to give with one of the other ones and the natural candidate is the R here that reflects its sign okay now first of all think about where this is happening this is happening at a distance or a coordinate R equals two mg over C squared C is a huge number C is a huge number and it's in the denominator this is a small distance this is a small distance but it's a small distance that depends on the mass it's a small distance that depends on the mass for the earth this distance would be about a centimeter okay so we don't have to worry about it and thinking about the gravitation of the earth it was hitting at odd direction is it ticking in a space those ticking its ticking in a time direction clocks clocks don't care about what coordinates you use we're going to discover that this is an artifact of a guff of particular coordinates there's not going on special at this point we're going to discover that this is just awkward coordinates that make it look like something's funny is happening here a switch of space and time there's nothing funny happening there and we're going to work out and see exactly why but for the moment it does look like there's something going on C squared R it does look like there's something going on when this thing vanishes when this thing vanishes here this thing becomes infinite it looks like some terrible thing is happening to the geometry in fact the geometry is completely smooth over here nothing special happening the light cone is perfectly healthy on all in all regions in here nothing funny going on but we have to work harder to see it this was the form that's what shield derived the Schwartzel the schwartz shield or fire history Einstein wrote down some field equations Swart shield and what Einstein knew about them is he basically knew this piece here and he knew some oh oh there's one thing that's important here one thing that's important yeah which I neglected to say how big is this correction here well if I'm if I want to estimate how big this correction is in nonrelativistic limit I want C to be very very large if C is very large then I should expand this this one over a small quantity this is a very small quantity 2 mg over C squared R alright and so this is approximately equal to 1 plus 2 mg over C squared R plus Corrections if you take 1 over 1 minus X 1 over 1 minus X is equal to 1 plus X plus x squared plus X cubed Tata Tata that's the binomial expansion and so the first correction if I write 1 over 1 minus 2 mg R is 1 plus the small quantity the next contribution is two mg over C squared R squared this is typically small if you're not if you're not too close to the center of the thing if you're at a reasonable distance away so the two mg over RC squared is very small the next terms of even smaller and smaller and smaller notice there are a 1 over C to the fourth one over C to the sixth one of the C to the 8th so what we're doing here is we're modifying this term here also with Corrections which are very small the next correction here does contribute to the motion of particles and so forth but not in the nonrelativistic limit the next correction here would be something of order 1 over C to the fourth the X square blah-blah-blah-blah-blah and 1 over C to the fourth is very very small so as long as you don't get too close where R is too small this correction is pretty negligible that's order 1 over C to the fourth this should be a C squared here I'm sorry this should this should be a C squared downstairs yeah even this term even this term is small even this term is small because there are two C squares this gets multiplied by another C squared so even this small too small to be important I don't think I've convinced you that I can throw this term away have I okay let's do it again this is one plus two mg over C squared our next term is two m 4m squared G squared over R squared one over C to the fourth dot-dot-dot and it all gets multiplied by 1 over C squared all gets multiplied by 1 over C squared and notice that even the first correction here is 1 over C to the fourth much smaller than the original term DT squared even these things are small but especially the correction to a small thing is even smaller so this is negligible compared to this and we simply don't have to worry about it for slow particles for particles moving much slower than the speed of light okay that that's a let's let's come back to this metric here before we analyze it and see what's going on at this point where you cross from space to time where you cross the horizon of the black hole let's use it let's use it and see if we can figure out how particles move not in the nonrelativistic limit but let's do better let's see if we can answer how particles move are without taking any approximations if we're not going to do any approximations incidentally we might as well set C equal to 1 let's set C equal to 1 keep in mind that we set C equal to 1 you can set C equal to 1 it's just a choice of units and now see if we can see how particles move in the presence of this metric and it's not so hard we can consider we can consider two kinds of orbits two kinds of orbits that I want to work out for you or at least I want to show you how they work circular orbits will find a little bit of a surprise about circular orbits and then radial orbits radial orbits maybe I'll do radial orbits I think we'll do radial orbits first I think we'll probably run out of time for circular orbits which would you rather see circular orbits what it would look good to both will do but one falling radially in it's not an orbit it's not an orbit it's right it's not an orbit just a crash a crash a crash here's our here's our object a rocket crashes into it and the question is the question was already asked how long does it take for that rocket to reach the horizon how long does it take for it to reach the horizon no we're going to watch we're going to talk now about the external observer the external observer who uses Tifa time whereas somebody falling in in the rocket uses the towel uses palatine proper time in the rocket or coordinate time from the outside somebody watching with their clock somebody on the outside watching this thing with their clock uses time T conventional time somebody falling in uses proper time okay so we could or we can ask both questions we and they have different answers very different answers okay now the main point of this is to show you how the things work so let's go back to the lagrangian for the infalling part for the particle where is it let's go back incidentally where you start with any such calculation like this you start with the Lagrangian for the particle then you work out the Euler Lagrange equations or whatever you need to do with it and and you then solve the equations there are tricks there are tricks you can do and I'm going to show you tricks that that allow you to do the problem okay the Lagrangian yeah the action is minus M will leave the C squared out we set C equal to one times the square root of D tau squared and the square root of D tau squared let's write the Tau squared over here is equal to here it is oh here it is right here d tau squared I'm also going to assume that the particle does not change its angle when it comes in it comes in along a particular angle and we will not have to worry about the piece of this over here we're not moving around an angle and so if we're not moving around an angle as the particle falls in we can say that D tau is equal to the square root of one minus two mg over R DT squared minus one over one minus two mg of our D R squared let's put this into here you this is the action in this integral here but not to make sense out of it I'm going to divide by DT squared and multiply BT T 1 minus this and now what is this quantity dr by dt squared that's just the velocity the radial velocity squared in other words r dot squared so this is r dot squared here and I have a Lagrangian is my Lagrangian L is equal to minus M square root of 1 minus two mg over R minus 1 over one minus two mg over R times R dot squared the coordinate in this case is honor and here we have a Lagrangian OK fortunately this problem is easy or at least it's easy to see what's happening let's not worry about the exact solutions exactly solving it is a little bit hard but seeing what happens is is easy always a conserved quantity what's it what's what's the cool what's the conserved quantity that's always there energy right energy is always there what is energy in terms of the Lagrangian energy in terms of the Lagrangian is the Hamiltonian so I'm going to remind you what the Hamiltonian is the Hamiltonian the first thing we do is we calculate the momentum conjugate to R what's the momentum conjugate to R that's partial of L with respect to r dot remember that partial du of L with respect to the velocity is the momentum conjugate to R and if we work that out let's see what we get that's not too hard the square roots of our an annoyance there are real annoyance but they're not the natural so terrible you know I don't think I will work them out I'll let you work them out partial of L with respect to r dot is a straight though a forward thing to do you just differentiate with respect to r dot and you get P sub R what do you do with it you multiply it by r dot and then you subtract the lagrangian partial of L with respect to r dot you multiply it by r dot you subtract the Lagrangian and that's called the Hamiltonian or the energy that's the energy of the system and you can work it out you work it out I'm not going to work it out here it is H is equal to one minus two mg over R times the mass divided by square root of one minus two mg over R in brackets times are dot squared I believe so minus one over one minus two mg over R R dot squared complicated thing it's an ugly thing it's a complicated thing but it's a definite thing and what does it depend on it depends on the velocity that depends on the velocity and that's the energy what do we know about the energy we know that it's conserved we know that it's conserved and so we can write that this is equal to e and that it doesn't change with time it does not change with time ok let's see what that says that tells us that we can solve for our dot we can find out what our dot is as a function of the energy I'm going to write down what our dot is as a function of the energy you just solve the r dot you solve this equation it's like this is much easier than it looks you just square it you square it you get rid of the square root to get an e squared on the left hand side you have the square of this here and you turn it over and calculate the R dot okay let's see if I can find it here it is here's what you get it's it's not at all complicated r dot is approximately equal to I'll tell you when it's the R minus two mg over oh it's easy no no it's not even approximate its exact our dot is equal to oh very easy yep all right that squared is equal to the energy squared times one minus two mg over R is that squared minus one minus two mg over R cubed divided by the square of the energy it doesn't really matter what it is it's a definite thing but what does matter is what it does when R is near to mg when R is near to mg something bad happens to it our dot is approximately equal to the square root of R minus 2 mg divided by 2 mg remember with two mg is mg is the radius of the horizon R equals to mg and what this is telling us is as the particle Falls toward the horizon its velocity goes to zero as it gets as it falls towards the Schwarzschild radius its velocity gets smaller and smaller instead of accelerating it gets smaller and smaller and smaller can we see that from here I wonder yeah the velocity gets smaller and smaller no don't know g20 okay so let's see what's going on here this goes to zero this goes to 0 as R goes to 2 mg this goes to zero so this is zero yeah this will go to zero the first term here also goes to zero and then we'll have in the denominator R dot squared or x one over this thing here square root so this goes to infinity this goes to infinity right so if our dot did not go to 0 this would be 0 over infinity which would be 0 but it can't be 0 it's supposed to be the energy which is conserved all right the energy is conserved and so our dot cannot sorry oh the only way to avoid to make this be 0 over 0 and make any sense which would be to have r dot go to 0 but here this is you could solve the equation solve it solve for r dot as a function of E and M and everything else and this is what you get so as our so as our goes to 2 mg R dot goes to 0 it's a surprise it's a surprise you might think the thing accelerates but instead it slows down slows down and slows down now can we see why this is it slows down gets asymptotically slower and slower and slower and never passes this point never passes that point just asymptotically approaching it takes forever to get to here can we see why this is I wonder I think we can let's go back to the we have it let's go back to the Schwarzschild metric okay the swatch of metric is the tau squared is equal to one minus two mg over R DT squared minus one over one minus two mg of our the R squared and let's forget the other piece of it the other piece of it is not playing any role now how does a radial light ray move a radial light ray which is moving in let's say toward the or for better yet moving away moving away let's think about a light ray moving away from the object at distance R either moving away or falling it it doesn't matter pulling in or moving away okay what is the light ray a light ray is a solution to the equation that the the interval here this little distance here is like like like like means d tau squared equal to zero and our original where is it our original light light means D tau squared is equal to zero so a light ray satisfies let's solve this we're going to set this equal to zero and that will tell us that D that will tell us that dr is equal to dt times one minus two mg over R now let's do it we're going to set this equal to zero now let's multiply both sides by one minus two M mg over R that will give us one minus two mg over R squared DT squared equals D R squared take the square root and now divided by DT that's a light ray that's the fastest thing that can ever be a light ray nothing can be faster than a light ray falling and this could be falling in the R by DT either positive or negative this would be an in falling light ray and what do we find that as R goes to the Schwarzschild radius what happens to this at the Schwarzschild radius this goes to zero this is equal to R minus two mg over R and that's equal to the radial velocity R dot the magnitude of the radial velocity even a light ray gets stuck as it's moving toward the is it moving with the speed of light of course it's moving with the speed of light what else kind of light ray do but the speed of light has this property that in these coordinates in this particular set of coordinates the speed of light goes to zero as you get closer and closer so nothing even including a light ray can get past this the surface here or so it seems or so it seems in fact that would have been a much eat this is a much easier way to think about it isn't it it's yeah what about when it's far away how about when it's far away when it's far away this one is the most important thing that's one so it's far away the Arditi is 1 1 here means the speed of light as it starts moving in closer and closer it eventually gets to the point where this goes to zero and then the Arditi simply goes to zero something odd happens beyond that point okay something odd happens beyond that point and we're going to work that out okay this was a I the problem with general relativity is the principles are pretty simple and the computations are always ugly they're almost always ugly this the this is nothing nice about the actual computations the principle simple and they give rise to complicated equations it's just the way it is but if you ask let's have some questions now and their fight goes into a black hole so radiation well obviously if it was right at that boundary wouldn't get anywhere Nina would never escape from the boundary yeah yeah this fact that this is the magnitude of the velocity the magnitude of the velocity goes to zero either for ingoing or outgoing radiation let's see what do we do here we had we had an equation that this is equal to the squared before I took the square root now there are two solutions when I take the square root that the two solutions when I take the square root of plus or minus the plus solution the Arditi is outgoing radiation all right outgoing radiation or outgoing outgoing light moving out the radial velocity is positive the negative solution here is light falling in but in either case the magnitude of the velocity goes to zero so you're right if you start a light ray right at the short childrenís it won't get anywhere in fact a light ray will just sit at the right at the Schwarzschild radius a light ray interior to the short Sholay Arabia will fall in a light ray exterior will go out but the closer it starts to the finish will radius the longer it will take to escape that's what this is saying well it materializes outside the black hole it doesn't materialize mm-hmm has to be a little bit outside but you know if it starts out very very close it will sit there for a very very long time and then after a very long time it will be observed to be a little bit away in a little more way a little more way and then will pop up go off for the speed of light water is falling velocities loss you it looks like 4,000 with nonzero stationary mark no no wise what do you think it has none zero mention anything it's apparently stationary we're going to find out that not that it that from the point of view of somebody falling with a photon the photon moves of the speed of light and everybody including the photon just go sailing right through this is an artifact of peculiar coordinates at the horizon there's nothing really going on that so that's the strange of the horizon the virtual radius is the horizon one other thing that I should emphasize is that if we were really talking about the Sun or the earth the earth well the earth is how big there's 12,000 kilometers and 6,000 kilometers in in radius yeah is 6000 kilometers in radius on earth this Schwarzschild radius for the earth is about a centimeter so it's a little thing like that and this means that you can't you don't trust the solution in the interior where there's real stuff this solution is only correct outside the region where the stuff is where the matter which creates the gravitational field is so this is the solution of the equations of motion for a lump of material outside the lump now it's only if the lump shrinks and becomes smaller than this radius here that this radius means anything particular it's only if the whole thing with the collapse down to a radius smaller than Schwartzel radius that the Schwarzschild rate that the that the solution is meaningful and correct down to such small distances so nothing there's nothing funny going on the other center of the earth in particular even this thing which I'm telling you is really not so funny even this this does not happen near the center of the earth the center of the earth does not have a horizon rather the solution is different in the interior than it is on the outside they have to be pieced together but if the earth were to collapse and shrink the smaller than the Schwarzschild radius and this would this would be the solution by mixing product from our point of view anything it falls into the black hole doesn't really go into just all the matter that falls it stays officers from us watching it in a sense that's trouble right we will we will not perceive it passing we will watch it then it will asymptotically get slower and slower and slower and slow down never passing it whereas somebody riding with the infalling material will just sail right through the horizon something could ever get to the veteran of a black hole of a call from sort of an outside observer could never grow and by struck netting backwards even never form so if you just accept whatever correct no no no this candidate from the perspective of an outside observer you accept nothing but these equations applaud cold would never get larger because everything front perspective an outside observer would asymptotically approach the event horizon but never cross it here that's true a black hole could not grow and by extrapolating backwards it couldn't even form yeah right this is what you're saying sounds right but it is wrong but to understand to understand how it's wrong you might say that a black hole can never get started you say let's suppose let's start with a little little tiny black hole yeah let's start with a tiny tiny black hole and now let's throw stuff into it and try to make it bigger that's what people tell us can happen the tiny black hole can grow you throw more planets into a black hole eventually it'll grow from being a solar mass size to being a million solar mass size so it grows but I've just told you also that nothing ever gets to the horizon so there's something not quite right hmm right okay we're gonna we're going to work out some examples we're going to see exactly how a black hole grows and the answer is it does grow as you're standing at this you're outside of Black Belt now you drop a sports car and compartment no and you track it with the Raider yeah you look at how fast it's yeah at least at first it starts going faster and faster yeah sure what though I'm sorry yes that's right yes certainly it starts going faster and faster but until it starts to go slower and slower at a point of point econo there's a point there's a specific point where so it turns over it starts to look slower and slower no well let's say the radar gun is out here right yeah so as this thing Falls it falls it flattens out it must flatten out because the two edges of it the the front of it in the back of it never get past the horizon and at the front in the back never get there past the horizon and both asymptotically move toward it it gets some it gets more and more squashed but now the radar gun sends in its radar and its radar does the same thing its radar never catches it the radar will never catch the infalling thing it will also slow 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 here they'll both be inching slower and slower and slower toward the horizon with a radar beam never quite catching the horizon yeah yeah yeah yeah I just take it'll take longer and longer that's right it'll take longer and longer when the radar beam bounces off it will be enormous Lee redshifted Yeah right exactly this is how ya know it's what I said about the radar beam not catching up would be true if this were falling in if this were a light ray but it just takes an enormous ly long time for the radar beam to catch up and when the radar beam reflects its enormous ly redshift good not good enough for a reason to catch you speeding close to the right question yeah so the statement of the nine-year statement I can tell you today that speed of light is constant no matter what front finger that only true in special relativity or when what we're saying here I guess know what what is true alright let's again we have a we have somebody falling in who's in a car and that car gets squashed down and down and down so we could ask the light ray which is crossing that car as the car falls in it gets more and more squashed and at the same time time slows down we haven't talked about the slowing down of time it's in here time slows down meaning to say when this quantity is the way to think about it this quantity is going to zero when this quantity goes to 0 a given amount of DT corresponds to a smaller and smaller and smaller proper time in the car so we'll get to it we'll work all this out but what does happen is the light rate when you take into account how the clocks behaved and how the meter sticks behave and everything the person in the car will see light go from the back of the car to the front of a car with the speed of light time the clock and the meter sticks and everything will expand contract whatever they have to do and in that car following with the car the light will go from the back to the front in exactly the amount of time that it's supposed to go so that's that's not going to change we'll work that out we'll work it out we'll work it out yeah yes good if gravity is by the shape of space-time what does it mean to unify gravity with the other forces what is the attempt even me anyway well is there some character of the other forces that are that are affecting the space most ideas about unifying gravity together with the other forces involves adding other dimensions to space when you add other dimensions to space the gravitational field becomes more complicated it has more components and the additional components of the gravitational field when viewed from three dimensions or three three space one time dimension the other components become the fields associated with electromagnetism associated with the standard model and so forth so from the point of view of three plus one ordinary three plus one dimensions three plus one dimensions means three dimensions of space one dimension of time you have a bunch of fields in addition to the gravitational field when they're put together into a higher dimensional space all you have is the gravitational field so that the that's most attempt at unification work along lines like that that the additional direction is a space motion in the different directions in the additional directions of space give rise to properties for particles for example which are their charges which are their various properties and that's where most the ideas of our unification go so in the end it's all gravity but gravity in higher dimensions you know in those theories that end up resulting in multiverse type ideas is the signature of the metric always one plus three and if you can ask in whether the current versions of the theory or you can ask whether in reality in reality nobody has the vaguest idea could every regions of space with odd properties of different signatures I have no idea so that sounds like a signature then it's mostly an empirical observation no it's more than that it's a classical it's a classical property of general relativity the general relativity equations of motion will resist a change in signature as the signature tries to get to the point where it's going to change the energies become explosive and and that's kind of like trying to exceed the speed of light energy blows up at the point where the speed of light the approached energy blows up as the signature of space-time stuff to get close to the point where it could shift so it's it's more than them it's more than a statement of empirical fact it's a property of Einstein's general theory on the other hand you can ask well what happens if you add to Einstein's general relativity quantum Corrections all sorts of things and then nobody knows thus far nothing anybody has ever seen in any version of quantum gravity string theory anything else has ever led to to a change in the signature does that mean it doesn't happen somewheres I don't know and I think that's boy way beyond us you beyond us certainly beyond me right now I'm about them you know probably humanists interest standpoint I've read that Shore shield was on the frontlines of World War one when he worked out those equations do you have any insight in it I mean given that they're so complicated how could someone do that what kind of back home or you have Oh your guess is as good as mine he was it he was said to be an astronomer but he was obviously a pretty sophisticated mathematician and yeah he was he was in in World War one I don't know it boggles the mind you know sometimes when I want to go into deep withdrawal I sit down and solve equations it is it is one way of going into withdrawal I can tell you for more please visit us at stanford.edu

Channel: Stanford

Views: 166,144

Keywords: physics, timelike, spacelike, lightlike, science, black hole, special relativity, speed of light, relativity, derivative, vector, space, time, eigen, mass, gravity, lagrangian, coordinate, surface, radius, orbit, velocity, stationary, growth, dimen

Id: quWN1V9jOf0

Length: 99min 7sec (5947 seconds)

Published: Tue Oct 30 2012