General Relativity Lecture 9

Stanford University we want to come now no we're not going to get deep into solving Einstein's field equations um they're awfully damn complicated if you were to sit there even to write them down explicitly is complicated if you really wanted to write them down as I said before the principles of general relativity are pretty simple but it's computationally nasty almost everything you try to do gets complicated fast there's a lot of coastal symbols I forget how many but a lot of them independent ones even more elements of the curvature tensor each Christoffel symbol has a bunch of dirt has derivatives the curvature tensor has more derivatives the equations get complicated hard to write on a single piece of paper the best way to solve them in fact the best way to even write them down is just to feed them into your computer in Mathematica and Mathematica will spill it will spit out answers whenever it can and on the other hand as I said basic principles are simple but going anywheres past the basic simple basic things tends to be computationally intensive so we won't do much computation we'll just kind of correspond we'll just concentrate on the meaning of the symbols and then I'll I'll tell you what happens when you try to solve them in various circumstances we may get a chance to do a little bit of solving in thinking about gravitational waves next week no gravitational waves this week just the equations okay so topic tonight is Einsteins field equations now before the we do that we should talk first of all about the corresponding Newtonian concepts so let's talk about Newton's version of Einstein field equations Newton didn't think of our fields he didn't have a concept of field equations but nevertheless there are field equations which are equivalent to Newton okay so first of all it's always a sort of two-way street that masses affect the gravitational field and the gravitational field affects the way masses move now John Wheeler had some way to say it which was a very clever I can't remember what it was you got it oh yeah something like that right always these two-way things so let's talk about the two-way street in the context of Newton first of all field effects particles right that's just a statement of F force on a particle gravitational force on a particle can be written as minus M times the gradient of the gravitational potential the gradient of the gravitational potential the gravitational potential I usually write as Phi and Phi is a function of position okay so everywhere is in space due to whatever to whatever reason everywhere is in space there is a gravitational potential called Phi of X and it varies from place to place you multiply it by M the mass of the object and you take the gradient and that tells you the force on the object that's a that's one aspect of field Phi of X tells particles how remove and in this case by telling them that what their acceleration should be if we write that this is equal to MA and the MS cancel and we just have the rule that acceleration is equal to minus the gradient of the gravitational potential okay so that's that's field tells particles how to move and on the other hand masses in space tell the gravitational field what to be alright the equation that tells the gravitational field what the B is the equation Pistone's equation does everybody know what those squared Phi means del squared Phi means the second partial of Phi with respect to x squared plus the second partial of Phi with respect to Y squared plus the same thing with Z let's call del squared Phi is equal to something which on the right hand side has the distribution of masses which are the things which are causing the field to be there all right so what is on the right hand side for pi that's a convention that's a convention well that's it's not completely a convention but there's a four pi but yet is a convention because another factor here is Newton's constant and if you don't like for PI's you can absorb them into Newton's constant they'll reappear in other places so don't think you're getting away with it by you're not putting the four pi here note is constant six point seven times ten to the minus 11th meters per blah blah whatever it is times one other thing the density of mass the density of mass is a function of position it's also a function of time masses can move around so on the right hand side we have mass densities sources on the left hand side we have the gravitational field Phi alright so we have these two aspects field tells particles how to move and mass particles in other words mass tells field how to curve well how to do whatever it is that it does you can solve this equation in particular in a special case in the special case where Rho prefer what is Rho mean Rho means the amount of mass per unit volume mass per volume in the case where Rho of X is concentrated let's call it a star doesn't have to be a star it could be a planet it could be a bowling ball but let's say a spherically symmetric object a completely spherically symmetric object of total mass m and it does not matter whether the mass is uniformly distributed in there meaning to say it has to be it has to be symmetric with respect to rotation but it could be more dense in the inside than the outside doesn't matter once you get on the outside of where there is any mass once you get out beyond the region where there is mass you can solve the equation uniquely and the solution of the equation is Phi is equal to minus the total mass times G the G is there because there's a G in the source equation here mass that's the integrated amount of total mass divided by our that solves this equation if you take this Phi which depends on position and you calculate del squared you will find that it's zero everywhere is outside the okay so this is a solution outside this is the outside solution I'm only I don't care about anything but the outside solution so I might as well shrink this to a point if I shrunk it down to a point then this solution would be valid everywhere except at that point the point has a total mass M all right so that would then be Newton's equations if I then plug this in no not sure constant just symmetric yeah not necessarily the mass density could go to zero right but it doesn't matter that doesn't matter okay the point is that if I take Phi written this way and then I take its gradient it just puts another are downstairs differentiating 1 over R with respect to radius which is what Dell does makes it 1 over R squared and winds up giving you F is equal to little m the mass of the particle Big M Newton's constant divided by R squared so it's all there and this is a field way the field kind of primitive field theoretic way to think about gravitation instead of action at a distance we have a gravitational field it's still really action at a distance because the Newton's theory if you move around the mass Phi instantly reacts to it and changes but it's a way of writing the theory making it look like a field theory okay this is the thing that we want to replace right here we want the replace by something which makes sense relativistically which makes sense in general relativity just to get a little bit of a handle to get started let's remember something about the schwarzschild geometry we pull the Schwartzel geometry out of a hat of course the point is that it's a solution of Einstein's equations this is Newton's equation it's a solution of Einstein's equations but let's just remember what we wrote then we wrote down a metric I'm not going to write the whole thing I'm just going to remind you what G naught naught was the time time component of the metric was one minus two mg over R oh I erased what the solution was didn't I let me put back the solution for a minute Phi equals minus mg over R okay now let's write Schwarzschild over here the only part of it that I'm interested in right now is just to remind you G naught naught is one minus two mg over R I've set the speed of light everywhere is equal to one as usual C is equal to one actually I don't need this equation and now I see that this required to the to the extent that that this can be identified that one minus two mg over R can be identified with anything over here we could think of this then as something like Dell squared let's see I'm let's get the sign right I always have trouble with the sign here I think it's del squared G not not is it minus or plus I have written down plus but I think I mean - because of this sign because of that sign here del squared of G not not G not not is just one - oh it's one plus Phi 1 plus 2 Phi it's equal to 1 plus 2 Phi 1 plus 2 Phi so del square G naught naught is just twice del squared Phi the one that has no derivatives derivative of 1 is 0 del square G naught naught is just twice del square root of Phi so del square G naught naught is equal not - 4 PI but the 8 pi G Rho now this should be taken with a grain of salt this this is just a mnemonic a mnemonic device to remember the relationship between some some aspects of general relativity and matter so already this begins to sound like matter or mass is affecting the geometry when we make this correspondence between Newton's Phi and Einstein's or Schwarz Shields metric we see that roughly it looks like I use the word roughly because we're going to be more precise but it has the rough texture of del squared of G naught naught is equal to 8 pi zero matter telling geometry how to curve so to speak okay all right but then of course this is not Feinstein's equations here Einstein's equations are a good deal more complicated than that oh the other half of the story of course is that in general relativity this equation over here is replaced by the statement that once you know the geometry once you know G naught not the rule is particles move on space-time geodesics so this equation becomes is replaced by the geodesic rule and the Newtonian field equation is replaced by something which I'm kind of naively just writing in this form we're going to do better we're going to figure out exactly well nice time figured out exactly what goes there okay before we do and before we write down the field equations we need to understand more about the right hand side the right hand side is the density of matter density of mass mass really means energy equals MC squared if we forget about C and set it equal to 1 then energy and mass are the same thing and so really what goes on the right hand side is energy density we need to understand more what kind of quantity in relativity energy density is it's part of a complex of things which includes more than just the energy density it's part of a complex in other words it's part of some kind of tensor whose other components have other meanings so let's go back and review quickly a little bit about the notion of conservation in this case conservation of energy and momentum in a simpler case that we're going to discuss in a moment conservation of charge conservation densities flows of things like charge and mass and let's just review a little bit that we've gone through before but let's do it again let me start with electric charge electric charge is simpler than energy for reasons we will come to electric charge the total electric charge of a system we can call Q that's a standard notation for electric charge Q I don't know where it comes from charge density in many situations charge density is called Rho but I don't want to confuse it with the energy density or the mass density over here so I'm going to give it another name I'm going to call a charge density Sigma and what is charge density charge density is you take a small volume a differential small volume you take the total amount of charge in there divided by the volume so it's charge per unit volume are in the limit of small volume differential volume okay and we'll call that Sigma Sigma equals schematically Q divided by volume at least has units of charge divided by volume we could draw this in another way here's if we draw time this way and space this way and draw a little element of space over here now that little element of space a little volume of space in this picture is 2-dimensional why is it 2-dimensional a little element of space because I didn't draw the third dimension that's all alright that's really three dimensional but it's too hard to draw it on the blackboard and then we have some charge which is moving around and passes into that little region there there's other charges out here which don't pass through it when I want to count the charge in a volume of space at a given instant of time it's almost like asking for the charges which pass through a little cubic area here which is now being represented as a little square area all right so it's charge per unit volume and you can draw it like this now next concept this is a Sigma the next concept is the flow of charge also called current now what we do is we take a little window let's forget this picture for a moment we take a little window in space this is a window in space now what is it's a window you know like a window our room doesn't have windows but I mean literally a small window it's not necessarily where the window is it's anyplace I want to put it okay it can orient itself in any direction the window is characterized by an area which I'll take to be infinitely small infinitesimal and orientation and a sense of direction through the window a window pointing that way is the opposite of a window pointing that way all right so this window is characterized by a little area and an orientation and angle and a scent so longer along it the current has to do with the amount of charge passing through that window per unit time per unit area windows have areas they don't have volumes all right but we also have to have a clock and allow the clock to proceed for a small amount of time in order to ask how much charge flows through that window per unit time charge J and in particular if the window is oriented along the X M axis X 1 X 2 X 3 or X Y & Z if the window is oriented along the x axis then J M is a vector which is equal to the charge through that window per unit area per unit time now in this case here Sigma was a charge divided by a product of three lengths volume here two charge divided by a product of two lengths times a time but in relativity time and space are nice and symmetric to each other so this dividing by area times time is again dividing by three three lengths one happens to be time like to happen to be space like but it has the same units of Sigma that's if if C is set equal to one if C is set equal to one and space and time have the same units and both of these can be thought of as charged through a window in this case the window is completely three space dimensions in this case the window is two space dimensions in one time dimension but they're similar creatures Sigma the charge density and the current of charge this is called current space current of charge those three those four things together form a four vector they form a four vector in the sense of relativity Sigma and J sub M together form a four vector J sub mu or some super moon with J zero being the charge density and J 1 J 2 and J 3 being the three components of the current good one more thing conservation of charge conservation of charge is a local idea what do I mean by saying it's local well conservation of charge could allow just sheer conservation could allow a charge this blackboard eraser if they have a little bit of charge on it might allow it to disappear over here and instantaneously appear over here in fact it could disappear over here and reappear at Alpha Centauri I always use Alpha Centauri is someplace which is so far away that doesn't matter and once we would and and that would mean that if that were possible you would say well charge is conserved but I would say who cares if charge is conserved if we can just disappear if we can just disappear arbitrarily to some very distant place it's just as good as saying it didn't wasn't conserved in my laboratory it just disappears charge doesn't disappear that way if it leaves the laboratory it passes through the walls of the laboratory that means it passes through windows that means it cannot leave the laboratory without a current flowing and that current has to flow out of the walls that idea is called continuity and there's an equation that goes with it the equation is the continuity equation if I take a little box and I look at all the charges passing out through the walls all the flow of charge is passing out through the walls if I'm interested in the charge per unit time that disappears out of the box let's say it's a unit box then the amount of charge in that unit box is just Sigma if the volume is equal to one and some units the charge per unit time that is leaving the box is minus Sigma dot why - because it's leaving the box if it's leaving the box Sigma is getting smaller that has to be equal to the sum of the currents passing out through the and buy some Gauss theorem or something that's just equal to the divergence of the current the same current here the divergence of the current is simply the total amount of current passing out of that small box divergence of the current inside here at the in in the vicinity of the box that's the current passing out and this says that Sigma dot is minus the divergence of the current we can also write this of course as we first of all we can write it as Sigma dot plus the divergence of a current is equal to zero Sigma dot means the derivative of Sigma with respect to time and this means the various components of the derivative of the current with respect to the corresponding direction of space is three it's a three vector curve is equal to zero and now if I write that Sigma is J naught this just becomes nice elegant that T T is X naught Sigma is J naught this just becomes a nice equation that derivative of JM j mu space-time index not just space index space time index J mu with respect to X mu is equal to zero incidentally I suppose if I want it to be really systemic I would put a sum over in here Sigma dot the Sigma by DT plus sum on em the j1 by DX 1 d j2 by DX 2 d j3 by DX 3 this just becomes the derivative of JMU with respect to X mu I've now used the summation convention here summation convention implied or implicit and same thing is equal to 0 so it becomes a nice tensor type equation J is a for vector X the four components of space and this has the nice look of a good equation the derivative of a tensor with respect to a position in curved coordinates in general if you had a thing like this in curved coordinates this would be correct in flat and no ordinary coordinates in curved coordinates you might replace this by the covariant derivative remember about covariant derivatives of tensors it turns out in this case it doesn't matter for charge currents it doesn't matter both in general it wouldn't matter when you go to curved coordinates you should replace all derivatives by covariant derivatives otherwise the equations are not good tensor equations now why do you want tensor equations you want tensor equations because you want them to be true in any set of coordinates all right so anyway that's the theory of electric charge flow current and the continuity equation this is called the continuity equation and the physics of it is that when charge either reappears it was sorry appears or disappears in a small volume is always traceable to currents flowing into or out through the boundaries of that region now let's come to energy and momentum energy and momentum are also conserved quantities they can be described in terms of density of energy density of momentum density of each component the momentum you can ask how much energy in the form of particles or whatever it happens to be including the MC squared part of the energy or you can ask how much energy is in a volume you can ask how much momentum is in a volume just look at all the particles within a volume and count up their momentum photons or electromagnetic radiation has both energy and momentum and that energy and momentum can be regarded as the integral of a density so in that sense each competes one of them each the energy and each component of the momentum are like the charge they're conserved they can flow an object or thing is moving then the momentum and energy may be flowing and the question is how do we represent the same set of ideas for energy of momentum now there's a difference between charge and energy and momentum electic charge is an invariant no matter how the charge is moving the charge of an electron is always the same the charge of an electron does not depend on its state of motion therefore charge itself is invariant charge is invariant the the density of charge and the current of charge are not invariance for example if I have a given charge and I look at in a different frame of reference here's my charge but I walk by it I have trouble doing that you know what I mean I walk by it with a certain velocity I think it did that pretty well didn't they yeah okay let's write it again yeah good certain amount of charge I look at that charge and because of Lorentz contraction I say that the volume of that charge is one thing you sitting still assign a different volume to it right one of us assigns a smaller volume let's say you assign a bigger volume I assign a smaller volume because I see it the Lorentz contracted if we take the charge and we divide it by the volume we will not agree about the value of the of the charge density but that's okay charge density is the component of a four-vector it's not an invariant charge density is a component of a four vector similarly you see charges well let's say yes you you see charges standing still and you say there is no current why the charges are at rest all of them I'm moving and I see a wind of charges passing me I say this current right we're both right of course charge density and charge current are not invariants they form together a four vector now energy and momentum are more complicated the total energy and momentum not the not the density of them but the total energy and momentum are not invariant I see a particle standing still the whole particle not that not that it's density I see a particle standing still and I say there's some energy there magnitude you're walking past it and you see not just the e equals mc-squared part of the energy but you also see kinetic energy of motion you're walking past the particle or the object sees more energy not because of any Lorentz contraction of the volume that it's in but just because the same object when you look at it has more energy than when I look at it the same is true of the total momentum not the flow not the not the density of it the same is true of momentum you see an object in motion you say there's momentum there I see the object at rest I say there's no momentum so energy and momentum unlike charge are not invariant they together form the components of a four vector and that four vector P mu includes the energy and the components of momentum p m where m labels of directions of space so each one of these is like a Q it's a conserved quantity each one of them is like a Q now energy can be in motion and we can ask or before it's in motion energy has a density associated with it energy has a density or incidentally e is also P not good all right come over to here for a minute density was the time component of J because it's a density we'll assign it a component zero zero for time but it's the time component of a density let's think about the time component of the energy P naught is the energy but let's think about the time component of its density not the end not the total energy but the time component of the density in other words how much energy is in a small volume not the total energy but energy within a small volume we're going to call that T north-north now where the notation T came from I don't know I know I don't know worried oh yes at some point in time it was tension but it's a long historical long historical evolution T naught naught now what are the two indices naught naught the first North indicates that we're talking about energy the second North indicates that we're talking about its density so you can think of this then as the time component meaning the density of a thing which is itself a time component namely the the energy T naught naught and it's a function of position if you integrated over position it tells you how much total energy you have but energy can also move energy can move from place to place and like momentum sorry like charge when energy disappears out of a region it does so because it passes out through the through the walls of the region and so energy also has a flow it's the amount of it's exactly the same idea the amount of energy passing through a window per unit time is the current of energy if you like we don't call it the current of energy we call it t naught 1 by the work T naught naught is density of energy and again I'll emphasize the fact that the density is one of these zeros I think it's this one and the fact that its energy is the first entry here next the flow of energy along the direction along the direction X 1 the amount of energy the amount of energy passing through a window oriented along the X 1 axis that's called T naught not because it's energy but then 1 because it's a flow along the X 1 direction likewise there's a T naught 2 and a T naught 3 these three together form the flow of energy and this is the density of energy exactly the same way as the continuity equation is derived the continuity equation for energy is derived and what does it say for the moment the first index here is just passive it just tells us to tells us what we're talking about we're talking about energy it's the 0 1 2 & 3 here which are like the components of the current here alright so what it tells us is that the covariant derivative with respect to X mu of T naught mu is equal to 0 this is the analogous continuity equation for energy but everything that I said about energy we could now say for any one of the components of the momentum so let's go to the components of the momentum now let's say the component pm the component PM also has a density put it up here component M R of momentum it also has a density and that density is called t1 well M T M naught naught here indicates that it's a density the M here indicates that we're talking about the M component of the momentum so we read this the density of the M component of momentum it's also a function of X and likewise we can consider the flow of the impotent of momentum M component of momentum flow along Direction n of X momentum is a conserved quantity it can flow each of its components can flow along some direction let me give you an example of some examples yeah component of that stress here in here that just means a function of position any possess means a function of position yeah it means XY and Z all of them function up position right all right so we can go a little further and we can say the same equation is true even if we replace energy by a component of momentum in other words we could replace naught by in but now I have an equation like this for all four possibilities we can just call this T mu mu for each new in other words new could be time in which case we're talking about energy or T could be space in which case we're talking about so we have basically what it comes down to is the flow and densities of energy and momentum form a tensor with two indices one index tells us who we're talking about energy of our momentum the other index tells us we were talking about density or flow and that's what and that is called the energy momentum tensor the energy momentum tensor whoops let's see um the energy-momentum tensor has an interesting property which i have not proved but for example take TM naught that is the density of the MS component of momentum compare it with T North M that's that's the flow of the energy this is flow of energy this is density of momentum it's a general property of relativistic systems which I'm not going to prove now which tells you that this matrix is symmetric TM naught is equal to t naught m we're not going to prove it it takes a little bit of work to prove it it's proved relativistic invariance allows you to connect this with this there's a theorem of relativistic mechanics that are relativistic field theory essentially all relativistic field theories the energy momentum tensor is symmetric so let's add that in and then we have the energy momentum tensor a big square matrix T naught 1 T naught 2 blah blah blah well T naught 3 that's all T 1 naught T 1 1 and so forth and so on we'll come back to the meaning of these elements in a little while this one is clear this is energy density these are fairly clear their flow of energy this one is momentum density the flow of momentum so they're pretty clear what they mean but we're going to find out that some of these elements have another meaning connected with pressure things like pressure things of that nature we'll come back to that in a while but at the moment the important idea is that the flow and density of energy and momentum are combined into an energy momentum tensor and each component of the energy oil the energy momentum tensor satisfies a continuity equation for continuity equations one for each type of stuff that we're talking about okay we'll come back to pressure a little while essentially a second rank or index of tensor just because it's not carrying the total energy Lewin is not a variant like total cars total energy total energy and momentum is non variant that's the first index I'm very OCD a charge and Earth yeah you got that the extra to get something out of that this mu here does not tell us what quantity we're talking about we know we're talking about electric charge okay so we could write this in another way we could say this is J charge mu okay I just put the Q up here to remind us what stuff we're talking about then the Mew over here tells us about the direction of flow when it's time like its density when its space like is flow so the second index is the thing which distinguishes flow along which axis the first index here tells us what quantity we're talking about same here T naught tells us we're talking about energy this M tells us we're talking about the flow of it along the axis M how much it was flowing out to a window orient along the M axis but fortunately we don't have to remember T naught m and T naught air to you know they're equal that's the symmetry we don't have to remember which or it's easy enough to remember that one of these must be one of these must correspond to the fact that its energy and the other that its flow but it's not important to remember which is which because they're interchangeable two indices here okay so now let's let's returned seething anger suspect exactly yes it is actually I'm not no that's right the entire metric of the entire matrix is symmetric correct that's correct we'll come back to its structure a little while maybe not tonight but the are and some of the meaning of its elements but for the moment what we've learned now is that the notion of energy density is incomplete it's part of a multiplet of things it's part of a the words tensor of course it's part of a tensor so the right-hand side of this equation is part of the tensor the left-hand side must also be part of a tensor but whenever you have a tensor equation you can't have a tensor equation that says some particular component of a tensor is equal to some other the same component of some other tensor let's say that then that is is an example let's take some particular component of a four vector a let's say a three it's a four vector and this is the third component of it and I assert that there's a law of physics that says that a 3 is equal to B 3 3 being the Z come by the Z direction does that make sense as a law of physics well it only makes sense as the law of physics if it is also true that a 2 equals B 2 and a 1 equals B 1 why is that why can't you just have a law that says that the third component of a vector along the z axis is equal to the third component of some other vector and not have that the other two components are equal it's a simple that that if if it is always true in every frame of reference that the third component of a is equal to the third component of B if it's true in every frame of reference then by rotating the frame of reference we can rotate a3 that we can rotate the third axis until it becomes the second axis and so if it's true in every frame of reference that a3 is equal to b3 then a2 must be equal to b2 and a 1 must be equal to b1 if it's to be true in every frame of reference that's an example of why equations need to be tensor equations of the form a sub M equals B sub M or vector equations full vector equations okay when you go to relativity the same thing is true even concluding the time component of equations if it would show for example in some frame of reference no in every frame of reference that a certain four vector now this is a four vector this is the time component of it is always equal to B zero in every frame of reference then the only way that can be true is if all components are equal in every frame of reference a mu is equal to B meal for the same reason Lorentz transformations are not so different from rotations and unless all the components are equal you'll always be able to find the frame of reference in which a naught will not be equal to B not unless all four components are equal so good laws of physics must be tensor laws of physics in particular if they're to be true in every frame of reference now here we have an equation that involves a right-hand side which is a particular component of some tensor it's a component and not not component of the of the energy momentum tensor let's not worry too much about whether this left hand side was just a was just a guess of what the left hand side might look like but the right hand side is the energy density and it's what you expect to be on the right hand side of Newton's equations but Newton's are sorry the right hand side of Newton's equations but the right hand side of in Stein's equations must involve not a particular component of a tensor but it must generalize to something that involves all components of the tensor so that means Einsteins generalization of Newton must read something like this the right hand side must be 8 pi G same a PI G times team you knew a special case being when mu and nu are both equal to time and then that becomes the energy density but if the equation is to be true in every frame it has to be a tensor equation what has to be on the left-hand side what has to be on the left-hand side must also be a tensor with two components a rank two tensor otherwise the equation doesn't make sense so on the left hand side must be some tensor with the same kind of tensor structure it must be symmetric because the right-hand side is symmetric and have whatever properties the right-hand side has but it's not something which is made up out of matter it's made up out of the metric it's something which is made up out of the metric it has to do with geometry and not with masses and sources so the left-hand side we will just say we'll call it capital G mu nu the only thing we know about it is it's made up out of the metric it probably has two derivatives in it to compare it with this here it's it will involve the metric in some form and very likely two derivatives the metric that's the kind of thing we would like to find to put on the left-hand side and when we find a good candidate for it we can then ask if we're in a situation where nonrelativistic physics should be a good approximation does this gene you knew reduce to just del square G not not perhaps it does perhaps it doesn't if it doesn't then we throw it away and try to find a different rule all right so we need to know what kind of thing G mu nu can be all right so let's so let's explore the possibilities G mu nu is a tensor made up out of out of the metric it has two derivatives or at least it must have some terms which have two derivatives to make it look like that so it's not just a metric by itself has to be a thing with two derivatives what kind of tensor can we make out of the metric and two derivatives alright so we've already talked about one it was the curvature tensor I mean remind you about the curvature tensor the curvature tensor was made up out of the Christoffel symbols now for our purposes tonight just writing down these equations as reminders you start with the Christoffel symbols and I'll just remind you what they look like this is equal to one-half G Sigma Delta times D tau D tau means D by DX tau G Delta Nu plus derivative with respect to new of G Delta Tau the first two terms are gotten by interchanging power new and then the last one is minus derivative of with respect to Delta I believe of G new tau now what's the only important thing right now this this involves the derivative of a G involves the first derivative of G one derivative alright next what about the curvature tensor now I'm going to write the curvature tensor down but the only important part of it is that it involves another derivative so here it is the curvature tensor you know on all its glory are mu nu tau Sigma and this is the thing which tells us that this real curvature if any component of this is nonzero anywhere in a region of space the space is curved okay so and what is equal to it's equal to I think D mu gamma nu Sigma Tau minus D nu gamma sigma mu tau and then there's another term with gamma nu lambda sigma gamma mu tau lambda don't I'm not sure there's any reason to be writing this down it's just the general overall structure which is interesting for reasons new lambda sigma gamma nu tau lambda summation convention assumed main point is the Christoffel symbol which is not a tensor has first two derivatives of the metric and the curvature tensor has first derivatives of the Christoffel symbols it also has things which are quadratic in the Christoffel symbol okay this means these terms here have second derivatives of the metric the derivatives of derivatives this term higher has also two derivatives squares of first derivatives so this is the kind of thing we like to see we like to see an R we like to see a tensor which involves two derivatives of the metric and that's it two derivatives of components of the metric that's a candidate or components of this various components of it are candidates to appear on the left hand side of this equation but wait all has four components the curvature tensor the Riemann curvature tensor has four components the left-hand side of this equation only is allowed to have two components why because the right-hand side has two components so this can't be the left-hand side of the equation itself what can you do to it to make a thing with two components you can make a thing with two components by contraction by contraction of components remember the rule if you set Sigma for example I'll tell you what happened you could set Sigma equal to tau in some you would get zero if you actually plug in what the curvature tensor is if you set Sigma equal to tau and sum you will get zero if you set Sigma equal to Nu and some you won't get Sigma you won't get zero you'll get something that we can call the tensor our new towel your is your problem can track you just because you get a covariant tensor where you need it contravariant this we can all exactly what I was coming to next okay right we can build a tensor from a tensor with four components you can build a tensor with two components by contracting indices but you have to be careful that you don't get zero you will get zero some of the sim trees of this thing the various minus signs that appear here in fact what's the various minus signs that appear here will wind up giving you zero when you contract Sigma with Tao they will not give you zero when you contract Sigma with nu or Sigma with mu but the two tensors you get by contracting Sigma with nu and symbol of mule happen to be the same tensor apart from a sign so there's really only one thing you can build it's a theorem the well-known theorem I don't know whose theorem it's called there's only one tensor that you can build out of two derivatives acting on the metric which has only two two indices it's called the Ricci tensor and it's a contraction of the Riemann tensor it has less information the Riemann tensor has a lot more components than the Ricci tensor it has less information meaning to say it doesn't in itself can be zero without the full riemann tensor being zero so this is called the Ricci tensor and as always if you have a tensor you can raise and lower its indices that means there's also a thing called arm you Tao there's also a thing called our new towel and also our new towel with both of them raised you can raise and lower indices using the metric tensor and so you asked you asked downtown you for each such tensor there's another one with upper indices another fact about the Ricci tensor is that it happens to be symmetric in particular are Miu tau equals R tau mule now that you just check by by using its definition I don't know and I don't know a simple quick argument about it all of these things are fairly complicated this Ricci tensor is symmetric just a property that has defined the way it is and so a possible left-hand side would be the Ricci tensor a question mark 8 pi G team you know now there's another one that you can make there's another one you can make another tensor that you can make so it was not unique only one other only one of the possibility and that's to begin by contracting mu and Tao lowering you can contract mu in Tao by multiplying this object is called R it's a scalar it's a scalar it's R mu mu which is also equal to G mu tau times R oops G Newtown arm you tell G mu tau the action of this G lowers the index towel and mix it into mu and then it becomes just exactly this thing here these are the same object this is called the curvature scalar it's a scalar it has no indices left at all so it's not what we want on the left hand side but we can multiply it by G mu nu multiplying it by G mu nu does give us a tensor that's another possibility I'm not recommending either of these at the moment I'm just saying from what we've said up till now either of these could be possible laws of gravitation they both involve second derivatives of the metric tensor equaling something on the right-hand side which looks like a density of energy and momentum okay which one shall we pick well we know one more thing we know one more thing and that's the conservation of energy and momentum or better yet the continuity equation for energy momentum if we believe that energy and momentum has the property that only disappears if it flows through walls of systems in other words if there's a flow then we are forced to the conclusion that Dinu covariant derivative of T mu nu is equal to zero that's the continuity equation D mu T mu nu is the continuity equation from which it follows that D mu G mu nu is equal to zero so the first thing we could do is we could check whether either of these two tensors satisfies this relationship if not then the left-hand side simply can't be the right-hand side unless we give up conservation look continuity of the energy and momentum I'll check this one for you let's just check this one and then I'll tell you what the other one does let's check let's calculate D mu of G mu nu R we want to put some parentheses around this okay the first fact is covariant derivatives satisfy the usual product law that this is equal to D mu G mu nu times R plus G mu nu Li mu R that's just a product rule for derivatives covariant derivatives are no exception this is true now what about the covariant derivative the metric tensor remember the covariant derivative the covariant derivative of the metric tensor is zero zero covariant that's where we started that's how we calculated the the Christoffel symbols by starting with the assumption that the covariant derivative of g is equal to zero and the reason for that is because covariant derivatives are by definition tensors which in the special good frame of reference are equal to ordinary derivatives but the ordinary derivative of g in the good frame of reference the good frame of reference being the frame of reference in which the derivatives of G of zero so the covariant derivative of G is equal to zero this term is not there and now R is a scalar the covariant derivative of a scalar is just the ordinary derivative scalars don't have any indices derivatives covariant derivatives of scalars are just ordinary derivatives when I write here R + don't run away yeah run away ok what about this this is just equal to G mu nu times the derivative of R well certainly in general the derivative of the curvature is not zero in general the derivative of curvature the curvature scalar if the curvature scalar was constant everywhere as we know that's not true we know that there can be geometries which are more curved in one place less curved in other places even flat in some places certainly it is not the case that the derivative of R is identically equal to 0 ok and the G mu nu here doesn't help you can look you can lower you can lower the G mu nu get rid of it then we'll just say that the derivative of R is equal to 0 so here's what we find first of all the covariant derivative the covariant derivative of this guy over here is not equal to 0 but it is equal to just G mu nu D mu R so it doesn't work no good bad what about this one well we do the same thing we calculate D mu of R mu nu and it's a little harder but not much a little bit harder I'll tell you what the answer is it's equal to one half G mu nu D mu R again it can't be 0 for the same reason that this one can't be and happens to be exactly one half the covariant derivative the corresponding covariant derivative problems be one here but now we know the answer if we take G mu nu R and subtract off one half R mu nu a better yet take R mu nu and subtract off I go is a 2 or 1/2 um okay so what do I know I know that our derivative derivative of R mu nu ya is equal to 1/2 yeah I think I got it right all right but I now have I now have a obvious thing to do we combine this with this arm you know with some coefficient times G mu nu and then we will get a thing whose appropriate derivative is equal to 0 so where's Jimmy lu g mu nu is our mu nu minus one half G mu nu are that's a theorem that there is nothing else made up out of two derivatives acting on the metric that has the property that it's covariantly conserved this would be called covariantly conserved it just is nothing else that the that so either oh I take it back of course you could have twice it or half of it or 17 times it but now it just becomes a question of matching this equation to Newton's equations in the appropriate approximations where everything is moving numb relativistically and one of two possibilities either there is some correct numerical multiple of this which matches this or there isn't if there isn't then we're in trouble then we're in trouble when I say it matches it we look at the time time component the time time component of this equation has row on the right-hand side it has T naught naught on the right-hand side T naught naught is row so we take this equation in the nonrelativistic limit everybody moving slowly not too strong a gravitational field we plug it in and we hope that with some appropriate numerical coefficient here this equation and this equation are the same in an appropriate the limit okay the answer is yes they are the same and the same with coefficient one numerical coefficient one just a piece of luck that it turned out with the coefficient one nothing nothing deep about that and this is what Einstein this was in Stein's calculation he knew what was going on pretty much but he didn't quite know what the right equation of motion was I believe in the beginning he actually did try arm you know evil to Tim you know and eventually found realized that it didn't work I don't know how many weeks of work it took him to do all of this but in the end he discovered capital G mu nu equals R mu nu minus 1/2 G mu nu R this is called the Einstein tensor this is the Ricci tensor and this is the curvature scalar so this is now known as Einstein's field equations and they do reproduce Newton in the appropriate limit but now we see something interesting we see that in general the source of the gravitational field is not just energy density but it can volve evolve energy flow it can involve momentum density and they can even involve momentum flow now as a rule the momentum flow or even the energy flow even the energy flow certainly the momentum flow but even the momentum density are much smaller than the energy density why do I say that it has to do with the speeds of light in the formulas if you do this but if you put the speeds of light into the formulas just like energy is always huge because it gets multiplied by C squared but on the other hand momentum is typically not small huge because it's just mass times velocity so velocity is slow if you're in the nonrelativistic situation when velocity is slow energy is big energy density is by far the biggest thing the other components of the energy momentum tensor are much smaller typically decreased by powers of the speed of light so in the various nonrelativistic situations the only thing that's important than the right-hand side in a frame of reference where the sources are moving slowly in a frame of reference where the sources are moving slowly the only important thing on the right hand side is Rho it's also true that in the same limit the only important thing on the left hand side is the second derivative of G of G not so with non relativistic limit these things match but if you're outside of the nonrelativistic limit things places where sources are moving rapidly or even places where the sources are made up out of particles which are moving rapidly even though the whole thing may be not moving so much other components of the energy momentum tensor do generate gravitation they do generate the curvature it's not just energy which are which or mass if you're every sometimes call it in the Newtonian mechanics so you can drive the continuity equation by analyzing little differential element of fluid or gas or them is there anything analogous is there an ounce way to come up with this I mean this set of C like G let's try this like that or third huh so it would is there some sort of a sickle way to come up with this as opposed to this equation yeah without using the continuity equation no to obtain equation well the logic here is a little bit different the logic is that the continuity equation basically comes from the idea of conservation and local conservation none of this crap where you have something disappear and reappear on Alpha Centauri that logic is just as good in relativity as it is in in nonrelativistic physics stuff having to pass out through the boundaries in order to disappear so in some sense that continuity equation is more fundamental from this point of view but there is another point of view which is the which we'll talk about maybe next time which is the action formulation which is much more beautiful and much more condensed where we introduce a principle of least action for the gravitational field and all of this just pops out hard calculations they're not easy but but it pops out getting the bullet Raja the curvature scalar that's all just the curvature scalar is the Lagrangian density okay the car sorry the curvature scalar plus some things for matter for the movement for the matter of for the for the sources yeah so the reasoning here seems to be over looking for is something that say cancer cancer I guess and that satisfies the continuity equation that in it it says strictly is you medical stuff right so could there be something else there is no it's known that there isn't it's taller that they're gone is that a physical well it's a mathematical fact but is it a physical thing well I can't think of any simple physical argument for it but but it is a fact I know it's an easy fact the only the only tensor this was known from Rima and the only tensor that you can build that involves only through two derivatives of the metric was the Riemann curvature tensor and things that descend from it by by contraction so yes Einstein knew that it had to be built up out of the curvature tensor and it's it's not hard to go through to exhaust all possible things that you can do with the curvature tensor essentially did it all possible things that you can do with the curvature tensor to make a thing with only two indices there's only two things you can do one of them is arm you know and the other and this you know this is not hard to prove it's very straightforward so Einstein presumably knew that that this particular Condor combination satisfied a continuity equation that was Einstein so we must have had to do a little bit of work to calculate this thing over here the mule arm you know this one he had to do a little bit of work to calculate that I assume that what he did was just plug in and it would take you about 15 or 20 minutes to do it as usual the principles are simple but though by the time you've manipulated all the indices and written down the Christoffel symbols and worried about getting the signs right and so forth it'll take you a good 15 minutes to get this this done so I can envision him having done this little calculation noticed that agreed with this calculation and that if he combined them in the right proportions that he would have what he needs there as I said there is another argument the other argument this is a much more elegant but this is was the first round of things that he did yeah two questions like this what is when you did that interaction tensor before indices of two indices the stuff that we had that thrown away oh yes oh yes oh yes oh yes and oh yes yeah yeah another way Yeah right okay so yeah well this is quite a bit of content to it but let me point that one thing that just as in Maxwell's equations there are solutions which don't involve sources all right you can have you can find solutions of Maxwell's equations that don't have any sources um or you can find solutions in regions of space in which there are no no no sources let's consider the case with either no sources or when we're in a region of space where there are no sources outside of all the sources then on the right hand side we have zero so let me just show you that there's a simplification in that case it's just a little simplification but but the equation does become simpler in that case let's suppose the team you knew is equal to zero in other words are mu nu minus one-half G mu nu R is equal to zero or better yet arm you know is equal to one half G mu nu R now let's calculate R by on the left hand side contracting nu and mu it's equivalent to lowering an index here we have to do the lowering on this side also and then setting nu equal to MU do you remember what G is with one upper and one lower index it's the kronecker delta go back to your notes if you don't remember the G with one in upper and one lower index which means Kronecker Delta ordinary Chronicle Delta in general relativity the chronica Delta is always considered to be a thing with one upper and one lower index but it means the same thing zero when mu is not equal to Nu one when it is equal to no it's the unit matrix the reason okay well let's just just remind ourselves of this all right so let's let's now contract mu with no that means set mu equal to Nu and sum what do I get on the left-hand side on the left-hand side they get R on the right-hand side I get 1/2 R times this object g mumu or better yet delta mu mu what is Delta mu mu 4 4 there are 4 pieces each each Delta Delta naught naught is 1 Delta 1 1 is 1 Delta 2 2 is 1 Delta 3 3 is 1 this is just 4 and we get the stupid equation that R is equal to 2r the only solution of that is that r is equal to 0 not the curvature tensor not even the Ricci tensor but just the curvature scalar that does not imply that the Ricci tensor is 0 it only implies what it implies which is the Ricci scalar is equal to 0 in which case in this special case when this is equal to 0 you can drop this term 1 half G mu nu R is just 0 Einstein's field equations become a little simpler in what is usually called the vacuum case the vacuum case means in regions of space where there are no sources in regions of space where there are no sources the whole Einstein set of Einstein equations is just the equations that are mu nu is equal to 0 the solutions are not trivial they contain gravitational waves with no sources just like there are electromagnetic waves but another example is the Schwarzschild metric now the Schwarzschild metric itself is analogous roughly speaking analogous to to a point mass outside the point mass there is no matter nothing so just like Newton's equations with Everywhere's outside the mass the equation is the same as it would be for just empty space the Schwarzschild metric is also one for which everywhere is outside the singularity the equation that satisfied is the is the vacuum Einstein equation so a simple thing it's not simple to do it's a it's a real nuisance to do but a conceptually simple thing to do is to take the Schwarzschild metric and calculate R mu nu and check that it's equal to zero you will find that it is if you take this watch field metric so it's a bunch of G mu news sit down and spend the rest of the day calculating Christoffel symbols and then curvature tensors and contracting them or put it on the Mathematica and you'll find out that it's exactly equal to zero now of course it's ambiguous at the singularity singularity everything is so crazy it doesn't make any sense but the components which are contracting become infinite they're sort of like the point mass to singular to undefined at the origin to have a value but everywhere is away from the singularity yes the Schwarzschild metric is what is called Ricci flat saying that this is equal to zero is sometimes called Ricci flat it is not the same as flat so gravitational waves satisfy this Schwartzel metric satisfies it accepted the singularity and that the that's the basic facts about the Einstein field equations I think we'll quit there yeah analysis ETA what's it again you do the same analysis that you did there a contracting yeah but you leave here you don't set it to zero you leave T oh okay good good yeah yeah yeah you can um right okay so you can do that let's let's let's do that yeah okay let's see what we get we have armed you knew doesn't matter whether it's upstairs or downstairs are armed you knew minus one half G mu nu R is equal to 8 pi G T Tim you know okay now what are we going to do we're going to contract we're going to contract these two indices this is going to give us R minus 2 R is equal to 8 pi G T mu mu I contract the index or are the same but what's the same thing T mu nu g mu nu okay so that now tells us let's say this is minus R minus R equals 8 pi G and let's just call it t t is by definition the scalar that you get when you contract the two indices here so what do we have we have R is equal to minus 8 pi G T and now we can put that back into the Einstein equations put this back we get R mu nu minus one half G mu nu times R which is plus 8 pi G T equals 8 pi G T mu nu and now let's take this thing here and shift it to the right this was a plus sign here huh so we get 8 pi G team you knew my 8 pi G multiplies team you new minus one half G mu nu T where T has gotten by contracting um right so so alright so here's what we get in other words Einstein field equations can be written with our muna on the left hand side but on the right hand side you have to compensate by subtracting one half G mu nu times T yeah it's called the trace of the energy momentum tensor and yeah the trace of the energy moment up okay I'll tell you when it's zero it's zero for radiation electromagnetic radiation it's zero when the mass of the when when the it's zero for massless particles like photons or gravitons it's for electromagnetic radiation it would be zero they would be zero for particles with mass the energy momentum tensor this thing is not zero so yeah can you this rockets about describe the physical meaning of the Ricci scalar the Ricci tensor the Riemann tensor Bobby this for ya Riemann tensor has to do with going around the little curve and seeing how what kind of you know what kind of what kind of rotation you do I don't know a simple answer to that I don't know any some particular physical significance to or geometric significance I'm sure there is but whatever it is is it's it's not really very transparent there are much simpler objects than the full Riemann tensor on the other hand that's it's kind of difficult to visualize their individual meaning so I'm gonna say no that I don't know a good simple way to think about these things but I'm leaning over direct reality more first Russians know right they are right oh right so you asked whether you lose information well the answer is yes you can have geometries where arm you knew is equal to 0 where the Riemann tensor is not equal to 0 and an example that we'll explore a little bit is gravitational waves gravitational waves just like electromagnetic waves they don't require any sources of course really we really expect that in the real world an electromagnetic wave is made by an antenna or something but a solutions of Maxwell's equations you can just have electromagnetic waves that just propagate from infinity to infinity and just no sources in the same way you can have gravitational waves which also have no sources those gravitational waves satisfy our mu nu equals 0 but they are most certainly not flat space there's all sorts of distortions of space going on so I think maybe a possible somewhat satisfying thing is to see what a geometry that has our mu nu equal to 0 but for which the curvature tensor itself is not equal to 0 and we'll explore a little bit what a gravitational wave looks like so that will help same thing of course is true of the Schwarzschild metric as long as you stay away from the singularity there's something there's real curvature there tidal forces all sorts of stuff but arm you know is equal to 0 so yes arm you knew has less information in it than in the curvature tensor it actually depends on the dimensionality in four dimensions there is less information in the Ricci tensor than in the Riemann tensor turns out in three dimensions the amount of information is the same you can write their you can write one in terms of the other either way and in two dimensions all of the information is in the scalar and taller is there's a scaler and you can make the other things out of it repair me so that was being there's some invariance by connecting the Ricci tensor which is scalar that phrases but there are combinations that you can you can subtract that information out of the agreement tents are not how to affect the gravitational field with that in gravity waves or that form are sorted by mother there's jitters there there's missing information and that missing information has no effect on the momentum energy tensor correct because we're even because the left hand is not gonna be intensive right so looks like so how is it going like an information manifest it means it means that for a given source there can be many many solutions many of which you know they all have the same left-hand side namely team you knew but they simply have different those different physical properties so for example the simplest situation is to say what if this is zero if it's zero does it mean that there's no gravitational no interesting geometry at all no it allows gravitational waves roughly speaking any energy momentum tensor you put there construct the solution and then add gravitational waves on top of it roughly speaking that's not exactly true but but it's roughly speaking true that any solution you can always add gravitational waves all right so the gravitational waves must be something that the contain more information than just a Ricci tensor than they do no fun factor ya know cuz well no little Cosmo know the cosmological constant can be thought of as part of team you knew yeah now we haven't we haven't said anything about a cosmological constant whether or not it's there the yeah okay from this point of view the cosmological constant can either be thought of as an energy momentum tensor or a component to the energy momentum tensor let's call it team u new cosmological for the cosmological constant would be some lambda times G mu nu so then if there was a cosmological constant you might write 8 pi G times lambda times a times G mu nu or this number here is just a constant let's just get rid of it and call it sum I guess it is really the cosmological constant that appears there you could it only depends on geometry a number of times geometry you can shift it to the other side of the equation and think of it as part of the geometry part of the geometric side of the equation or you can leave it on the right side and think of it as part of the energy momentum so the cosmological constant BIR can be thought of it either way did I cite use the equation you derive tonight to calculate the orbit of mercury which was then observed by astronomers and so that equation was was you do you know of a source that would show how that calculation was done good question let me see if I can find one um it's probably and I'm in one of Einstein's papers all of which I have at home and I'll see if I can find it yeah well I know how it was done you you want to see it actually done though it was done by by he didn't have the Schwarzschild solution but he did have the approximation to the Schwarzschild solution at a fairly large distance now the stood the the surface of the Sun is way way out at a distance much larger than the Schwarzschild radius of the Sun of course the Sun is not a black hole but outside outside the solar radius the geometry is exactly the same as the Schwarzschild geometry he didn't have the Schwarzschild geometry but he had a pretty good approximation to it that the you know a hundred times larger distance the Schwarzschild radius it's it's oh it's more than one hundred thousand million times larger than the Schwarzschild radius so he knew how to make a good approximation that was true far from the far from the center of a gravitating object and then he just solved orbits in that then after that geometry and again in an approximation the thing that allows you to do that is the fact that the Sun is is so much bigger than the Schwarzschild radius and that means the corrections from Newton are small you can think of it as just small corrections from Newton and work it out in in a kind of perturbation theory perturbation theory means just small corrections to something you already know yeah so most likely what he did was just take the Keplerian orbits for a light ray you know just there was a nice straight line and do a little bit of perturbation theory and then work it out you know again you asked me about the mercury or the so the bending of light I forget what you asked me mercury yeah so it undoubtedly just took the capillary in ordinary capillary in orbit took the Newtonian solution plus a small correction and fit the small correction on the left-hand side to the small correction on the right-hand side and but then schwa chilled immediately I think within a year of maybe less calculated his exact solution of the equations and from there you could do much much better than yeah okay you know that pretty much the the ordinary one over R squared potential low force law is the only power law force law which allows the orbits to be closed without a precession of the of the orbit it's a curious fact and it's somewhat accidental for more please visit us at stanford.edu

Channel: Stanford

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Keywords: physics, science, newton, force, einstein, field equation, mass, gravity, general relativity

Id: umdC_KD458g

Length: 104min 24sec (6264 seconds)

Published: Tue Dec 11 2012