5. Einstein's Field Equations | MIT 8.224 Exploring Black Holes

okay good evening welcome to the 8 to 20 for exploring black holes seminar I'm professor ed Bertschinger and I'm leading the seminar this evening on a subject of the Einstein field equations before I talk about the equations I want to take a step backwards to a higher level and just ask a rhetorical question of what is general relativity what is general relativity all about we've been studying the metric but is if you had to describe general relativity to your parents how would you describe it what is it what is it what does it attempt to do speak up Kevin it is that a question it really is a question how would how would you so what would you how would you describe general relativity in one sentence a simple sentence very perfect that's exactly what I wrote down it is a theory of gravity now the next question only a little bit harder what is gravity what is gravity someone else very good gravity is the force of attraction between two bodies we could refine that and say it's the force that's proportional to the products of the masses and inversely proportional to the square of the distance but the essential word there is force gravity is a force yet if you look on the back of the exploring black holes book the paradigm for gravity in general relativity is often quite different gravity is space-time curvature whatever that means matter tells space-time how to curve space-time tells matter how to move that's a very pithy summary but it doesn't sound very much like the Newtonian world that general relativity is also supposed to describe so what I want to do then is to just begin the process of describing how we talk about gravity first as a force then as space-time curvature and we'll make a comparison of the Newtonian perspective of gravity as a force the relativistic description of gravity as space-time curvature that will give us the foundation needed to state at least in concept the Einstein field equations and say what they describe actually many physicists have the impression the mistaken impression that general relativity so changed our view of the world that gravity is no longer a force it is only space-time curvature I think that is a mistaken view point just like in quantum mechanics where one can think of matter as being both particle and wave or light as being both particle and wave it's helpful I feel to think of gravity as being on the and a force and on the other hand a manifestation of space-time curvature with those with that complimentary perspective you'll be able to understand more of the physical content and meaning of the Einstein field equations so let me then remind you of the viewpoint of gravity as a force and we'll see that this viewpoint coming from Newtonian mechanics can be carried over to general relativity in Newtonian mechanics we write that mass times acceleration is a force Newton's second law I'm going to put a subscript I on this end because this is called the inertial mass when discussing gravity there is a mass factor on the right hand side that could be completely different although for some reason Nature has chosen that it be the same m times g this is the gravitational s Galileo some 400 years ago established the equivalence of these two masses this in fact is the foundation for Einsteins equivalence principle a very key ingredient in his theory of relativity and in the derivation and comprehension of the Einstein field equations this G vector it's a vector field every point in space has a gravity field produced by the sum of all the masses in the universe and let's see now we can ask what is this gravity field how do we calculate it we back up and remind you the content of general relativity is that matter tells space-time how to curve space-time tells matter how to move in Newtonian terms those statements would be mass tells the gravity field what to be the gravity field tells mass how to accelerate a vector whose magnitude gives the magnitude of acceleration whose Direction gives the direction of acceleration for any freely falling body we'll look at the corresponding equation in in relativity in a minute the other half of the Newtonian description is an equation which says how to calculate G well we remember the inverse square law with attraction so if we have a mass I'm going to put an M subscript I because I may have a collection of masses I'll have a test point X and then mass points positions X I if I take all the masses in the world calculate the vectors from X I to X then I can write the gravitational force as a sum over the masses with X vector minus X I divided by the magnitude of X vector minus X I cubed remember what this ratio does is to take the unit vector in the direction from actually from X to X I and then divide it by the square of the distance the unit vector is just the magnitude of is just the vector divided by 1 power of its magnitude that's why there's a 3 in this denominator instead of a 2 ok good enough let me see if I can use a little bit of this board down here I want to write this another way because when I make the connection with relativity it will turn out that there is another object called the gravitational potential it's much closer to the key ingredients in general relativity mic if this is not visible to the camera please bang on the glass and I will lower the board's in Newtonian gravity the gravity field vector field G can be written as minus the gradient operator of this function Phi and what that means is minus a vector whose components are the partial derivatives of Phi with respect to the coordinates so another way of describing Newtonian gravity is to say that there is a potential function Phi obeying a certain equation and I'll write down in a moment whose partial derivatives give the acceleration and that equation is if you like the Newtonian law of gravitation it will be the counterpart of the Einstein field equations I can write it in two forms an integral form so the integral form is like taking a sum over masses but I will replace that sum over discreet masses by an integral let's try a sex like this so this will be a minus G times the integral of Rho of X prime D three X prime over the distance X tax prime the integral has within it the mass density times volume that's just the mass that's contained in a little volume DX prime D Y prime DZ prime if in this formula I had replaced M sub I by exactly Rho of X prime times DX prime D Y prime Z prime and made the sum an integral over continuous mass field I'd have exactly the gradient of this function Phi I won't show that but it's not difficult to take the derivatives and actually work out the components and verify it this is the integral form unfortunately the Einstein equations do not have a comparable integral form solution in general they have only a differential form of the equations so if I want to compare with Newtonian gravity I must compare with the Newtonian differential equation for the potential you may not have seen it before unless you read closely some of the supplementary notes but the treatment here is very similar to that in 802 freshmen mechanics where the Maxwell equations are first presented in an integral form and well maybe only in an integral form unless you take the more advanced version or see a later version of electromagnetism there you'll find that the electric field can be gotten from a derivative of an electric potential where that potential obeys almost the same equation as the gravitational potential this is the differential equation for the gravitational potential and just to explain the Laplace operator that is shorthand for the sons of second partial derivatives acting on the potential now this is a lot of math I could try to describe gravity by saying everything is an inverse square law but I would not be doing justice to general relativity because in general relativity everything is not an inverse square law in fact everything is given by an equation that is deceptively simple I'll write down the equation for you it's a lot shorter than the Maxwell equations but it's content is far deeper and richer and in order to try and give you some explanation of the content of the Einstein field equations I'm going to be making analogies with Newtonian equations okay so now I want to take the second point of view of gravity as space-time curvature this is the standard way of thinking about general relativity I will show you actually that general relativity can regard gravity as a force as well but the foundations of the theory are based much more on the geometry of space-time than they are on gravity as a force the basic law of motion here I'm going to do this in two parts let me just review what I did here I said matter tells space-time how to curve space-time tells curved space-time tells matter how to move in the Newtonian case I started out with the second half of that sentence describing how matter moves in response to a gravitational field and the answer is that the path a particle takes X vector is a function of time obeys a second-order differential equation just the acceleration is equal to the local value of gravity computed at that point by summing up the gravity fields of all the masses the first part in how to calculate gravity comes from this sum or from the gradient of the gravitational potential now with gravity of space-time curvature I'm again going to start out with the second half of the sentence and ask how does curved space-time tell matter how to move how does matter move in the presence of gravity and the statement is this you'll have seen it in the notes I think for last week maybe maybe this week third set of notes freely falling bodies move along spacetime geodesics that is paths or else a curves of extremal proper time without any equations at all here is the law of physics that corresponds to F equals MA now we have to introduce a little bit of equations a little bit of understanding to describe this the concept of a geodesic the concept of extremely proper time this is simply the extremal aging condition and so what is this exactly mean well a curve is a path which I will write in this odd way x superscript mu as a function of lambda this hit this mu is an index that can equal zero one two or three it is not an exponent I don't mean X to the zeroth power or X to the first power rather X zero is one of the space-time coordinates in fact usually it's the T coordinate x1 if we're working with the Schwartzel metric might be the r coordinate and so on so this describes the path that a particle takes in four-dimensional space-time where lambda varies along the curve it is analogous in the Newtonian case to giving the position of the particle as a function of time now however time is one of the coordinates itself so we can't use time as the variable parameterizing occur okay a geodesic is nothing more or less than a curve of extremal space-time distance and in this case the proper time can be written in terms of this curve in the following way I apologize for the long equation you'll find that in the third set of supplemental notes posted at peak at the website I believe that something like this shows up on one of the problem sets maybe maybe the one due this week I don't recall anyway this is a way of writing the proper time that it takes to go from point A to point B in space-time along a curve the curve is a path parameterize path so I have point a point B and I have some path from A to B I could imagine many different paths that go from A to B the statement of physics is that of all these paths the freely falling body will take the extremal path for massive particle that in fact means the path of maximal proper time which may be the path of shortest spatial distance or it may not there's some path from A to B part of it takes now how does that one actually calculate that path I've got something that is an integral there Tao the statement is that if I vary the path look at all possible paths the one that has the longest proper time is the correct path that the particle will take this does not look very much like a differential equation there is no differential equation here but it implies a differential equation as the notes for this week explain I won't write down the equations in all their generality I will simply give you the result of applying the recipe presented in the notes to this function and finding the path of extreme old proper time it will look not so different from the Newtonian equation d squared X DT squared the acceleration equals G vector well let's see what what the equation is and then we can compare it oh I'll have to explain some notation in a moment okay so some notation here first of all I didn't really say what this G mu nu is the G mu nu are the coefficients of the metric G 0 0 DX 0 squared plus G 0 1 DX 0 DX 1 plus and so on you actually can work that out from taking the differential of tau on doing the integral and then squaring so there are 16 numbers in here that can be arranged into a 4 by 4 matrix called the matrix of metric coefficients or simply the metric okay that's the first bit of notation in an expression like this I'm taking partial derivatives of that metric with respect to one of the four space-time coordinates that's a little odd to say the least but that's what general relativity gives us then I've done another thing in this equation I've not bothered to write down some summation symbols implied outside of this would be a sum from mu equals 0 to 3 a sum from alpha equals 0 to 3 and a sum from beta equals 0 to 3 these are implicit because of the repeating upper and lower indices mu alpha and beta it's just that if you wrote down all the summation symbols every equation in general relativity we get about twice as long so when Einstein first wrote down this equation all about 1916 or so 1919 14 actually before he got the fuel equations he already introduced what we call the Einstein summation notation that just doesn't bother writing down these summation symbols so here's some long complicated expression but it is very much like F equals MA if we remember that the acceleration is the second derivative of the position that's analogous to here second derivative of the now space time position using proper time instead of bookkeeper time here it's bookkeeper time in Newtonian mechanics and on the other side of the equation instead of the derivatives of the gravitational potential in Newtonian mechanics there are derivatives of the metric know what is this equation really say well it says I explained here that if gravity is the only force acting on a body that body will take a path of maximal proper time going from event a to event B and it really is maximum if it's a massive body if I look at events that are not quite coincident suppose I start up I ask what's the path that a particle takes for a nearby event and then maybe goes to some other final event like so adjacent geodesics these are paths of extremal space-time distance in a curved space-time geodesics may start out parallel but then diverge or they may start out parallel and converge for example on a sphere suppose that you start on the equator at two points that are nearby on the equator but not coincident and you take the straightest possible paths that you can for each trajectory going due north and everyone who would agree that due north if you just follow due north you'll end up at the North Pole and that's the straightest path that you could take there's no shorter distance between us and the North Pole then by going due north two paths at the equator that start out parallel converge and meet at the North Pole every single due north trajectory converges at the North Pole so an indicator of space-time curvature is this property that geodesics which are initially parallel may converge or diverge parallel lines may intersect or diverge that shows space-time curvature in the language of components of position this equation looks rather different in this language it is analogous to F equals MA the only difference being that the gravitational force here is given by this complicated expression that depends on derivatives of the metric and so on any questions so far yeah these partial derivatives are components and of like a gradient I have to say kind of like because the gradient has certain properties that these partial derivatives may not have so I've described half of general relativity now I've told you how curved space-time tells matter how to move all that's left is the other half how does mass tell space-time how to curve and the answer is the Einstein field equations so I'm going to move this board up and I'll write the Einstein field equations you'll see they are extremely brief first ask the question how to space-time curve in response to matter or ask another way what determines the metric in the Newtonian case we have the Newtonian laws of gravity they determine the gravitational potential there must be some analog in generality and this is the path that Einstein to it he made the analogy he was forced to modify his thinking about space and time because he realized early on that a proper description of gravity would require curved space-time it's not so easy to think about curvature of space and time the words are longer than the equations themselves those are the Einstein field equations this capital G is Newton's constant I'm going against the philosophy of exploring black holes book by including Newton's constant but I do it for comparison with Newtonian equation of gravity where we retain Newton's constant G because that's how we learned it there's another object here G with superscripts it has nothing to do with Newton's constant it's an object called the Einstein tensor I'll explain a little bit about that in a moment and then eight pi we all know what that is although I should tell you that when Einstein first worked with this equation he wasn't by any means sure whether there should be an a pi there or a 4pi or something else that was part of his discovery to figure that coefficient and the final object on the right hand side this team you knew is called the stress-energy momentum ten sir well these are just names what do they mean what is a tensor that's not so easy to explain let me start by comparing just F equals MA in Newtonian gravity with the corresponding equation of general relativity the way that you learned F equals MA you used a vector sign over you call it a position vector the gravity vector really though what this equation means is a set of three equations for each of the components of the vector so I could write it in components as DX superscript I and d squared X by DT squared is equal to G I where I is equal to one two or three those would be called the components of F equals MA the mass is again canceled because of the Galilean equivalents of inertial and gravitational masses now this Newtonian equation is very similar to the relativistic equation at least the right-hand side is almost the left-hand side is almost the same I've now got four components because time is itself part of space-time we don't attempt to write separate equations for the space and time components they are together in space-time and it is the proper time along the path that describes the elapsed time instead of the bookkeeper time when I have a set of four objects like this four components in general relativity we could call those the elements of a vector in fact you call it a four vector because of the four four components a four-vector is a tensor with one index tensors can have more than one index when they have two indices that means a collection of four squared objects 16 objects the metric itself is a tensor this collection of objects G mu nu the metric coefficients they form a tensor and so sometimes we talk about the metric tensor other times we talk about the metric as being this equation with the coefficients multiplying the DT squared and the dr squared and so forth two aspects of the same thing so this boxed equation is actually 16 equations written in a very compact way still it doesn't look much like the Newtonian equations it is though completely analogous to the differential form of the gravity equation plus ones equation the only thing different is that factor of two on the right hand side and this object called T mu nu so I want to talk for a few minutes about the left-hand side and the right-hand side of this equation and see what they mean actually before I do having seen the summit the Einstein equations I want to look down in the valley as it were and ask about the content and nature of the Einstein field equations I started asking the question what is general relativity the answer is it's a theory of gravity well now the question would be what kind of theory of gravity is it's not only an equation a theory with tensors in it it's a very special theory of gravity so Feinstein a long time to figure out this theory and it's taken us decades to test the theory so you shouldn't take for granted this is not some kind of a mathematical axiom it's a lot of physics and a lot of physics ought to be motivated it ought to be tested or not to be loved we should identify the requirements for this theory now most of these requirements are self-evident you think about them very quickly so I'm going to I'd like to have a suggestion from the audience of one requirement of any requirement that a theory of gravity gravity ought to satisfied okay I'm going to call that requirement number two reduce to Newtonian gravity for weak fields and slow motions and they call that number two because there's a number one that's even more basic and important I think and every one of you could state it I'm sure what is the requirement for any successful theory of physics it has to match experiment we'll put another way it must pass all experimental tests now that statement is not vacuous first of all it implies that the tests are possible at an even more fundamental level it requires that the theory be capable of making tests of making predictions for tests excuse me the theory must be capable of making predictions that's something that is very easily forgotten by people I get lots of letters in the mail I'm sure dr. Taylor does too from people with all kinds of crank theories of the universe there are many people who think Einstein is wrong some of them may be listening on the tape and the very first requirement that I ask of any theory is that it make predictions that are capable of being tested now general relativity does and in fact some of those have to do with the Newtonian limit but you see this is even more fundamental because for strong fields gravity also sure past experimental tests what other requirements should a theory of gravity give you him satisfy a theory of gravity that followed special relativity why is the theory called general relativity I think we had a question in the recitation a week and a half ago about this what is a concise statement of the relativity principle the laws of physics must be the same in all inertial frames that's true in special relativity and in general relativity in general relativity inertial frames are only local they're not global and so the relativity principle becomes a statement that locally general relativity must satisfy special relativity all the laws of special relativity hold there are a few other points that are a little subtle let me motivate one by asking in Newtonian mechanics now I race to race one board I'm looking for you could remember this what's the source of gravity what causes gravity in Newtonian mechanics yes now what is what's the most famous equation of physics attributed to Einstein equals MC squared mass and energy are quote now in all the rest of physics including in all of special relativity different forms of energy are interconvertible into each other and into mass chemical energy nuclear energy kinetic energy how about gravitational energy well let's ask about the inverse-square law of electrical attraction when I make a hydrogen atom I bring together a proton and a neutron into a bound state is the energy of those of that system the same as the energy of an electron and a proton that are separated widely no because there is potential energy in your Tony in terms and that potential energy is negative for an attractive potential what if you just replace the electrical attraction by gravitational attraction the only thing that changes is the magnitude of attraction would the mass energy of that very weakly bound gravitational atom be the same as the mass energy of the separated particles well better not be because then I couldn't conserve energy I know that to pull apart two particles that are gravitationally bound to one another takes work and I could get that work by for example converting a little bit of mat rest mass into energy so the inter conversion of mass and energy implies that gravity gravitational fields themselves must have energy well if energy through equals MC squared is the source for gravity and gravitational fields themselves must be a source for their own existence some gravitational fields can induce or modify gravitational fields the field itself has energy now in this way gravity is quite different from electricity electricity is caused by electric charge electric fields carried by photons have no electric charge gravity is fundamentally different from electricity I'm often asked the question electricity is an inverse square law so how come I can't just take the Maxwell equations and then replace all the electric fields by gravity fields and I've got a relativistic theory of gravity well there are a lot of reasons why you can't do that well one of them is this fact that gravity itself carries its own source energy yeah the technically this means that the equations are nonlinear linear superposition does not work that's a technicality an important one is still technicality okay last plane I will write down an important one for Einstein's thinking has again to do with energy the field equations or the theory must be consistent with energy momentum conservation this really is an aspect of point number three locally the theory must obey special relativity and in special relativity energy and momentum can be moved from place to place they can be converted energy can be converted into different forms but energy and momentum cannot be created or destroyed without compensation so this is an aspect of point number three but it's an important one it was in Einstein's development of the theory he used it as a guide to successful theory okay I'm coming to a close got it probably a couple more boards I want to talk about the content of the Einstein field equations by explaining the left hand side and the right hand side just give you an example a little bit of an understanding of what they are of what what physics goes in here and I think I will start with the right hand side this team you knew what is this well it's a thing called the stress energy momentum tensor because it has three basic parts to it one is called what has to do with energy second has to do with momentum and the third has to do with something called stress in relativity first if I carry this Newtonian weak field concept I know that in a certain limit the source for gravity should be the mass density Rho but for consistency with relativity all forms of energy not just rest mass should be a source of gravity so this team unu includes a thing called energy density u and for example this could equal Rho C squared for a non relativistic gas gas with particles Rho would be the mass density C squared gives me conversion of mass to energy now how come the Einstein equation isn't simply that the laplacian of phi is four pi g you that would be the obvious way of promoting mass to energy and going from Newtonian mechanics to relativity therefore well the answer has to do with point number three locally satisfy special relativity energy density is not a Lorentz scalar value depends on your frame of reference energy is not a Lorentz invariant now Newtonian mechanics doesn't worry about this because there's no concept of the finite speed of light and no need to consider Lorentz transformations but relativity has those concepts energy is part of what's called the energy momentum four-vector whose elements are the energy and the momentum of a particle so far so good but this energy density is not the same as energy it is energy per unit volume and volume itself is not Lorentz invariant why is that if I take a little volume DX dy DZ a little cube put put little sticks to bound this and I view this in a moving frame I ask what volume does this cube occupy in a moving frame if I make a Lorentz transformation in the X direction I have Lorentz contractions so in a moving frame if the momentum equals zero in frame s then in s prime you prime is equal to gamma squared times u where this gamma is the famous Lorentz factor one over the square root of one minus beta squared it ends up getting two powers of this gamma the one power from the energy another power from the Lorentz contraction of the volume in fact you'll see why that happens Lorentz contraction means that this length goes to DX divided by gamma and it's in the denominator the presence of two factors of gamma this Lorentz transformation factor is what tells us that density energy density is part of a to index tensor the stress energy momentum tensor two indices two powers of gamma and in fact the rest of the components of this just right here in a local Lorentz frame this matrix team you knew it's a 4x4 matrix I can split it up into pieces where the zero zero component this would be 0 1 2 3 0 1 2 3 labeling the rows and columns of the matrix the 0 0 component is the energy density this component is the momentum density the 3 components along the top row and also along the column momentum density and then there's a 3 by 3 part in the middle that is called the stress tensor just to give you a little flavor momentum density that's easy to understand that's momentum per unit volume it says that if you throw a piece of chalk on account of its motion it has a gravitational field not the same as the Newtonian one but a moving mass creates a contribution to gravity much the same way by the way that a moving charge creates an electromagnetic field in fact you know that magnetic fields can be thought of as electric fields transformed from the rest frame of the particle the components of this stress tensor include very importantly pressure so 3 by 3 matrix whose diagonal elements are pressure why should pressure appear here well look at the energy density that's energy per unit volume I'm going to make just a little dimensional analysis and show you that pressure has the same units as energy per unit volume energy is force times distance volume is area times distance distance is canceled and we get force per unit area which are the same units as pressure now I have to admit this is not a very convincing argument but at least dimensionally it makes sense that pressure could appear in this equation in fact there's a more basic reason why pressure has to appear in this equation it's point number five together energy momentum and pressure participate in the requirements for energy conservation because as you learn in thermodynamics if you compress a box of gas you do work on it you apply a pressure you increase the energy in that box by pressure times the change in volume P Delta V work so in order to enforce energy momentum conservation and relativity it is a mathematical requirement that the source for gravity if it includes energy also include pressure now this leads to some amazing things for example it is possible to have substances with negative pressure actually if you take a rubber band an elastic band and you pull on it you have to do work to expand that rubber band whereas you take an ordinary box of gas you let it expand it does work on you releasing a rubber band that's been stretched releases some energy compressing a box of gas takes energy if the substance has negative pressure it creates a very bizarre gravitational field in fact if the pressure is sufficiently negative the gravity can be repulsive we think that may be why the expansion of universe is accelerating it's filled with a substance of negative sure now why there's a substance of negative pressure is a very very big question in cosmology and we'll be coming to that in several weeks in its class last thing come back to this board I've told you about the right hand side just a very brief description of the left hand side bring up this board this Einstein tensor happens to be let me see if I can get this right is up to trivial redefinitions the only tensor that is linear in the second partial derivatives of the metric are fishin's you need to take a course in relativity or differential geometry to see that in fact second derivatives of a metric tell you about the curvature of that space just like you may have learned in calculus that second derivatives of a curve tell you about the curvature of the path first derivative tells you the slope second derivative tells you about the curvature here you see the second derivatives of the metric in fact I have to say it's linear in the second derivatives and contains no higher derivatives I have to I'm not sure that I have all the precise state all the precise conditions on the metric here I'm doing this from memory I may not even need this second condition but that linearity means that in the Newtonian limit this G zero zero goes to see if I can get this right it'll be two times partial squared Phi partial x squared plus partial squared Phi partial Y square fact let me not use I'm sorry I just want it this equation is so so nice how this works out G 0 0 goes to partial squared of the metric coefficient G 0 0 with respect to x squared plus partial squared F G 0 0 with respect to Y squared plus partial squared G 0 0 with respect to Z squared and I'll check the coefficient in a moment in this weak field or Newtonian limit and I will check the coefficient by requiring that I recover the Newtonian equations of gravity so this is supposed to equal and I can see there's going to have to be a one-half one well no let me just plunge ahead 8 pi G times T 0 0 and depending on my sine of the metric that will be 8 pi G times u which is equal to 8 pi G times Rho if I pick units where C squared is equal to 1 and I'm going to have to check the sign here this can reduce --is to the Newtonian equation if I'm going to adopt the sign conventions of the textbook ebh and put plus signs here then g00 should equal to 1 plus 2 times the Newtonian potential and does it work out right if I take the second partial derivatives of this object the set of partial derivatives I have here in these parentheses this really is the laplacian of g 0 0 and i will get twice laplacian of phi equaling 8 pi G Rho and sure enough divide that 8 by 2 I get 4 pi 0 in the weak field limit Einstein field equations exactly reproduce the Newtonian equation for the gravitational potential well they do so with the implication with the cost at the time part of the metric this is the thing that multiplies the DT squared is not one but it's 1 plus 2 Phi and that means that time is warped go back down here detail squared is what is plus one plus two times the potential times DT squared plus other terms this is the week field limit of general relativity okay so it's a funny story with relativity all of Newtonian gravity that we're familiar with comes from warped time from the time factor in the metric all the popular discussion of black holes is associated with the warp with the curved space and these funnel and bending diagrams and you know we spent so much time already with the spatial curvature it's the time part of the metric that tells bodies how to fall in a gravitational field and that's what the Einstein equations are all about I've gone a little bit over so I think what we'll do is stop the formal presentation here but I'll stick around for any questions that you have and thanks a lot for being attentive and I'll see you on Thursday in recitation but if you do have any questions please stick around you

Channel: MIT OpenCourseWare

Views: 171,751

Rating: 4.8978434 out of 5

Keywords: mit, opencourseware, astrophysics, cosmology, GPS, big-bang, gravity, energy, momentum, trajectories, light, particles, orbits

Id: 8MWNs7Wfk84

Length: 69min 10sec (4150 seconds)

Published: Wed May 21 2008