General Relativity | Lecture 9

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General Relativity | Lecture 9
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(November 26, 2012) Leonard Susskind derives the Einstein field equations of general relativity and demonstrates how they equate spacetime curvature as expressed by the Einstein tensor, with the energy and momentum within that spacetime as expressed by the stress-energy tensor. This series is the fourth installment of a six-quarter series that explore the foundations of modern physics. In this quarter Susskind focuses on Einstein's General Theory of Relativity.
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were not going to dig deep into solving Einstein's field equations long dead awfully damn complicated if you want to sit them even tho right and down explicitly is complicated if you really want to write a Obama as I said before the principles of general relativity up pretty simple but computationally nasty almost Everything is trying to do gets complicated fast lot of estoppel symbols I forget how many years a lot of them independent ones even more are elements of the curvature tensor each Gestapo symbol has a bunch of derivatives the curvature tensor has more derivatives equations get complicated want a ride on a single piece of paper the best way to solve that in fact the best way even write that just a feed them into your computer Mathematica mathematical struggles about inches when Everett can and their but never him as I said the basic principles a simple but going anywhere as has 2 basic simple Of the basic things tends to be computationally intensive so it won't do much computational just come correspond with this term concentrate on me a meeting of the symbols and then pop tell you what happens when you try solve the various circumstances when they get a chance to do a little bit of solving them and thinking about gravitational waves next week no gravitational waves this week just equations OK so the topic tonight is Einstein's field equations before the we do that we should talk 1st of all about the corresponding Newtonian concepts so let's talk about no version of Einstein's field equations Newton didn't think about fields he didn't have a concept of field equations but nevertheless the off-field equations which are equivalent me OK so 1st of all it's always a sort of two-way street that effective gravitational field and the gravitational field affects the way masses moles are John Wheeler had some way to say it which was very clever can remember what was at stake gotta gap of aircraft that always to weigh things so let's talk about the way street in the context of nude 1st of all feel the effects particles as just a statement up and for summer particle gravitational force a particle can be written as minors and not times the gradient of the gravitational potential gradient of the gravitational potential gravitational potential I usually write fight and fight is a function of position OK so everywhere in space duel to whatever whatever reason everywhere space a gravitational potential coal-fired X and varies from place to place you multiply it by In the mess of the object and take the gradient and that tells you before strongly object that's Our maps 1 aspect of field fire wrecks those particles how removed and in this case by telling him the ability of acceleration should be free right that this is Eagle Dan canceled we just have the rule that acceleration is Jagoda minus the gradient of the gravitational potential that field those particles are removed and the other hand massacres in space color gravitational field work to beat the equation that tells a gravitational field with the baby is the equation plus owns equation Is everybody nor those squared fired means funny the means the 2nd part of fight with respect to X squared plus the 2nd partial fiery 8 with respect Y squared plus the same thing busy try Gold bills fight it is equal something which I'm a ranking inside has the distribution of massacres which are the things which are causing the field to be there at stores inside pie that's a convention at a convention our work it's not completely prevention of the floor price yet a convention figure some other factor here Newton's constant and if you don't like 4 pies you could absorb them into milk is constant reappear in other places so something to getting away with it by not before idea of his constant 6 . 7 times 10 to the minus 11 meters per blah blah however it is kind 1 of the things the density the density of masses of function a position it's also a function of time dresses canola around storm a right-hand side we have Mass densities sources on the left hand side we have the gravitational field 5 restore aspects field tells particles have moved and particles of awards may have tells field how to curb what I would do whatever it is that does yukos solace equation in particular in a special case in the special case where will was Ramiro means the amount the mass per unit volume Mass volume in a case where role of X is concentrated let's call a star of enough to be a starter planted could be a bowling ball but let's say a spherically symmetric object completely spherically symmetric object of total mess and that it does not matter whether the masses uniformly distributed in their meaning to say there has to be there has to be symmetric with respect the rotation but it could be more dentists in the inside and the outside
was matter once you get on outside of where is an e-mail I want to get out beyond the region where there is Maris you can solve the equation uniquely and the solution of the equation is 5 is equal come minus the cocoa mass the Times G G is there because Syzygy in the source equation here integrated amount of COBOL less divided by a R that solve this equation if you take this fight which depends on position and you calculate those square you will find that Zero everywhere outside view of this is a list outside this is the outside solution I'm only I don't care about anything but outside solution ask about over shrink list point if I shrunk it down to a point in this solution would be valid everywhere except at that point point has a total mess but that would then be Newton's equation to prevent club this road Castle is a mess number must share custody symmetric not dumbest showing the mist city could go to 0 the but does not represent nite socket for a visit therefore I think far-right written this way and then I take its Grady and it just put some of ah downstairs differentiating 1 or more armed with respect to radius which is what they'll does makes it 1 of our squared m lined up giving you bath is equal little and the rest of the particle begin Newton's constant divided by our squared with all that and this is a field way field come primitive field theoretic to think about the gravitation Mr. action at a distance we have a gravitational field still really action at a distance because in the theory if you move around a man's instantly reacts to it and changes but it's a way of writing the theory making it look like a field theory OK this is the thing that we want to replace right here want the replaced by something which makes sense relativistically which makes sense in general relativity just to get a little better handle to get started but remember something about this geometry we we pulled his fortune geometry out of a hat because the point is that it's a solution of Einstein's equations is no quick solution of Einstein's equations but that's just remember what we wrote then we wrote them a metric direct the whole thing I'm just going to remind you what Gino what what is the kind time component of that prick was warm minus 2 energy all art I erased with the solution was didn't die Of but that the solution from the flight equals minors R a picked out by over here the only part of it that I'm interested in right now is is just to remind you gene or not is 1 minus 2 energy also are I said the speed of light everywhere is equal to 1 as usual she is equal 1 that don't need this equation and now I see that this seek led to the heir to the extent that the this can be identified that 1 might storm GOR can be identified with anything over here we could think of this something like Dell squared let's see did the starring right there with her label would starring figures don't square gene or not is a minor subplots I have written them plus but I think I mean my is because of the signs because of that sign here don't square the G nor nor are or not it is just 1 minus poets 1 plus 5 1 plus 2 5 people don't want us to fight 1 cost of flight so don't square gene is just twice still squared 5 the 1 that has no direct remove 1 0 0 0 square gene or not His just twice those square defined Seville Square gene or not is equals not perform but that 8 GE well now that should be taken with a grain of salt this is just the new minor mnemonic device that work to remember relationship between some some aspects of general relativity and there not so already this begins to sound light matter or maps is affecting the geometry or make this correspondence between Newton's 580 and Einstein's which watch yields on Meta we see that roughly it looks like I used the word roughly because we would be more precise but has a rough texture of Dell square of gene nor is equal to party draw matter telling geometry how the curves of speak there of course this is not Einstein's equations Einstein's equations are a good deal more complicated than all the other half of the story of course is that in general relativity this equation over here is replaced by the statement that wants to know the geometry wants you know gene or not the rule particles move on space-time dude their 6 so this equation becomes is replaced by the geodesic rural knew Tommy and field equation is replaced by something which
I'm naively just writing in this war we were the better or figure out exactly what I stand figured out exactly what goes there OK before we do it fully right down the field equations we need to understand more about the the right hand inside the right hand side the density Of matter density of the Mets may have really means energy equals M C squared if we have to forget about seeds said equal to 1 that energy and mass of the same things and so really what goes on the right side is energy density we need to understand what what kind of quantity in relativity energy
density ears pot of a complex of things which includes more than just the energy density as part of a complex of words are part of some kind of cancer was other components have other meanings was 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 there discuss moment conservation of charge conservation cities flows over some of the things like charge mass Ames's review a bit that we can't go before the votes to again but me start with electric charge elected charges simpler than energy reasons newcomer electric charge the total electric charge the system we can call it's a standard notation for electric charge q wherever nor comes from charged games City in many S situations charge density it is called role but I don't want to confuse it with the energy density of the mass density over here somewhere give another Maine where color charge density Sigma was charge density charge densities you take a small volume the differential small volume you take a problem that the charge and they had divided by the volume source choice for you the volumes are in women of small different from bargain OK I'll call us Sigma Sigma equals schematic Q divided by Bali Greece as units of charts the barber bar we could draw the motorway here if we drop time this way and space this way the draw little element of space over here now that will element of space little volume of space in this pictures two-dimensional why is it two-dimensional a little element of space because they didn't go the 3rd dimension that's all right that's really three-dimensional bike trips toward a a blackboard really have a charge which is around and Paris Intel that that there's a other charges that he was not pass direct when I want to kill the charge in a volume of space at a given instant of time it's almost like asking for the charges which passed through will move cu every year which is now being represented as square earlier but charge per unit volume and you could drop like next concept this is a signal the next concept Is the flow of charge also called current now we do is we take a little window let's forget this picture from moment we take a little we know in space this is a window in space now Bob was that so we know little window on those have Windows early literally a small window that's not necessarily with window is to replace I wanna put it thinks it can orient itself in any direction the window was characterized by an area which appeared to be infinitely small and the personal in orientation and a sense of direction through the window or window pointing that worry is the opposite of a window pointing that were of the
window is characterized by a little area and orientation angle and they send Salonga along the current has to do with amount of charge passing through that window per unit time per unit area windows have areas they don't have volumes but we also have the havoc laughter 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 JET and in particular if the know it oriented along X access X 1 X 2 X 3 X Y and Z if the window is already in the along X Texas JE is back there which is equal to the charge through that window per unit area poll unit time now in this case here said was a charge divided by a product of 3 links volume here to charge divided by a product of Tulane transit time but relativity time and space and symmetric to each other so this dividing by area times time again dividing by three-year 3 Lakes 1 happens to be prime like a weapon to be space like but it has the same units a stigma that if if he is recalled 1 if C ascetical the 1 space and time have the same units and Botha is commit thought his
charge store window in this case the window is completely 3 space dimensions in this case the windows 2 space dimensions and 1 time dimension but this similar creatures said charge density and the current the choices is called current space current the charge those those 4 things together form a 4th a form of 4 vector the sense of relativity Cigna and J sub and together form of 4 vector Jason meal Hey for some superb with J 0 being a charge density J 1 J NJ 3 being the freak components Over Cora 3rd but 1 more thing conservation of charge concentration of charge is a local idea numerous local well concentration of charge could allow just sheer conservation could allow a charged black worries may have overcharge I'm about to disappear over here and instantaneously appear over here in fact they could disappear over here and reappeared office and Dorrie Trailways use office Peru's someplace which is so far away that doesn't matter and once we would and then that would need their fellow possible you would say what charges consumers but I would say shares of charges conservative in just disappear just disappear arbitrarily to some very distant place it it's just as good as saying it was observed in my laboratory it is disappears charged them disappear that way if leaves the laboratory it passes through the walls of the laboratory that means passes through windows and that means it cannot leave the laboratory without a current flowing and current passed flow the walls that idea was called continuity and is an equation that goes with it the equations the country no you crazy it may take a little box of all the charges passing out through the walls although flow of charges passing through the walls of and I'm interested in the charge per unit time disappears out of the box but say it's a unit box the amount of charging that unit box is just segment if the volume is equal to 1 and simulates the charge per unit time and that is leading the box is mightiest about why miners because leaving the box is leaving the box stigma is getting smaller that has to be equal some of the current passing out of the box and by some gals theorem or something that's just equal to the divergent Of the current the same currently at the divergent so the current is simply the problem of the current testing of small box divergence of a current inside her at the year's end in the vicinity of the box after that the current testing out and this says that stigma bot is minus the the divergence of the current Wigan also right of course as we could before so we can write about plus the attitudes of the current is equal to 0 simmer about means the derivative of signal with respect to kind and this means the various components Of the derivative all of current with respect to 0 the corresponding directions that is really a three-day physical 2 0 now if I write that similar is Jane ought becomes nice L again that T T is X Norwalk segment as Jane ought this just becomes the mice equations that derivative of J & J new space index not just based space time index view with respect to x new physical 2 0 incidentally I suppose if I wanted to be a really systematic I would put some of it in here certain about the signal by BT plus someone J won by X 1 PGA Tour by excluding J 3 by 6 3 is just becomes a derivative of Jamie with respect X new now used the summation convention here are some recent convention imply or implicit strengthening is equal to 0 becomes a nice cancer type equation are Jake is a format don't Texas for components was based endeavor this has the mice look over a good equation the derivative of cancer with respect to to world position I think curved coordinates in general if you have a thing like this In curved coordinates this would be corrected flat in the ordinary course curved coordinates you might replace this body we call variant derivatives Avacoll area derivatives of cancer Was that do not this case it doesn't matter for charged currents it doesn't matter but in general it would matter when you go to curved coordinates you should replace older readers by coal variant derivatives otherwise be equations on mark good equation whitey want considered Cuevas you want tensor glazes because you want them to be true in any set of Quarterdeck's oil that's the theory electric charge flow current and the continuity equation this is called the continuity equation and the physics of it is that when charge reappeared in area appears or disappears in a small volume is always traceable to current flowing into or out after the boundaries of that region now Contel energy and momentum energy and momentum are also conserved quantities so they can be described in terms of density of energy density of momentum density of each component of momentum you could
ask how much energy did form particles or whatever happens to be including the M C squared part of the energy by just how much energy is volume you could ask how much momentum as volume does particles with volume count of moment our photons or electromagnetic radiation has both energy and momentum of them and that energy and momentum commute regarded as our feet In the growth of and so that's since each come each 1 of them each energy and each composed of the momentum are like the charge that could serve think of flow of foreign object or other things moving the momentum and energy may be flowing and the question is how we represent the same set of ideas for energy momentum now there's a difference between charge and energy and momentum electric charge is invariant no matter how the charges moving the charge of an electron was always the same if the charge of electron does not depend on the state of motion therefore charge itself is invariant charms and variant be column density of charge and the current the charge are not invariants for example if I have a given charge and I look at the different frame of reference to is my charge but I walk by have trouble doing otherwise I walked by with certain velocity thing do that pretty well in the yard of charity now certain that a chance I look at that charge them because of the of the laments contraction I say that the volume of that charge is 1 thing you sitting still assigned a different Volume 1 of us assigns a small volume of assign a bigger volume I assigned a small volume because I see it the warrants contracted if we take the charge we divided by the volume we will not agree about the value of of the charge density but that's OK charged density is a component of a former vector it's not in invariant charge density it is a component of a what similarly you'll see charges I guess you see charges standing still and you say there is no current why the charges arrest all of I'm moving I see a window charges passing me I say this kind will both right of course charge density and charged current I'm not invariants they formed together a 4 vector now energy and momentum are more complicated the couple energy and momentum not does not the density of them for that helpless energy and momentum are not in very high C a particle standing still All particle knocked and not the the see a particle standing still and I say this summer energy services that you are walking past that and you'll see not just the E equals M C squared part of the energy but you also see kinetic energy emotions you walking past the particle or the object season more energy not because of any learns contraction of the volume that but just because the same object when you look at it as more energy the when I looked at it the it is true of the couple momentum not flow that that they have not be density of it the same is true of momentum you see an object motion you his momentum may I see the object at rest I say there's no momentum so energy and momentum on like charge are not invariant they together form the components of a 4th them and ask for vector key new clothes the energy and the components of momentum P M where an labels of directions of space are each 1 of these it is like a Q it's a conserved quantities each 1 of them is like a Q now energy can be emotion and we can't ask what be Florida motion energy has a density associated with energy has a density 0 incidentally E it is also keen not that I cannot be affirmed and this city it was the kind component of Jake because its intensity will assign it component 0 0 4 times the current component of the density let's think about the time component of the energy clean orders the energy but but let's think about the time component of its density not enough to topple energy but the time component of the density and other words how much energy nears In a small volume at Dublin GA but did you with a small or the lacked T. Nought Nought no way they notation T came from I don't know why not the old guests at some point in time there was tension long historical long historical revolution or now what is the 2 indices nor nor the 1st not indicates that we're talking about energy the 2nd not indicates that we're talking about its density so you could think of this as time component meaning the density of a thing which is itself upon compiled namely the energy T not not us affront to the position if you ain't did waited all the position it tells you how much total energy you have but energy could also moved Rijeka move from place to place in like momentum sorry like charge when energy disappears out region it does so because of passes out through the year the walls of the region and energy also has a flow see amount and that's exactly the same idea the amount of energy passing through window per unit time is the current of energy if you I will don't call it the current of energy we call it T not 1 or or God is
dead city of energy and again also emphasized the fact that density as 1 of these euros with this warm and the fact that energy is the 1st entry here next the flow of energy along the direction along the direction X warned the amount of energy care the amount of energy passing through a window oriented along the x 1 axis that's called T naught Nought because its energy but then won because it's a flow along the X 1 direction likewise visiting tool and a T North 3 these 3 together form a flow of energy and this is the density of energy the the exactly the same way as the continuity equation is derived the continuity equation for energy is derived was at the moment the 1st index uses passive just houses to tells us what we're talking about what talking about energy at 0 1 2 1 3 year which I like we components of the current year not so it tells us is that cold area derivatives with respect still like text new T not new is equal to 0 this is analogous continuity equation for energy but everything I set about energy we could now say Click any 1 of the components of the momentum so let's go to the components of a momentum now what say the components P. M the component also has a density appear component Air my oral momentum youth it also has a density and that density is called T warned well able to see them not here indicates that 2 density in here indicates that was talking about B and component of the moment so we read the density of the end component of momentum but also a function of X and likewise waking consider the flow of the and component of momentum and component the momentum floor along direction in acts momentum as it considers quantity it can flow each of its components comply along some direction however given example examples it or not on SAS really we we can best here in here I just means a function of position any position is a function of position yet his ex-wife and and function of position 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 and other words we could replace the not bite In but now I have an equation like there's for all 4 possibilities we could just call it key no meals for each new no adds new could be kind in which case with talking about energy or she could be space in which case with talking about our so we have is basically what it comes down to is 50 full and densities of energy and momentum formed a cancer with 2 indices 1 index tells us who would talking about energy of meant that the other index tells us we talking about density or flock and that's what matters called the energy momentum tensor energy momentum tensor the whoops the if 4th but Surrey armed the energy momentum tensor has an interest in property which I have not proved but there are for example take tea and want that is a density Of the M's composed the momentum to compare it with nor mats that's the flow of the energy this is full of energy this is density of momentum it's a general property of relativistic systems which I'm not going to prove which tells you that this matrix a symmetric nor what is equal to T North and we're not going to prove it bombed it takes a little bit of work to promote its proved relativistic invariant allows you to connect with various in the visit theorem of relativistic mechanics borrowed field theory essentially all road the field theories the energy momentum tensor summer so let's and that and then we have about the energy momentum tensor big square matrix T lot won t tool but Bob peanut 3 that's all she 1 Norwalk CT 1 1 and so forth so I will come back the meaning of these elements in a little while this 1 is clear this is energy density these are fairly clear flow of energy this 1 momentum density the full momentum so the pretty clear what they mean booger find out that there are some of these elements have other meaning connected with pressure things like pressure of things of that nature will come back a wire but at the moment for idea a floor and density of energy and momentum are combined into an energy momentum tensor and each component of the engine block the energy momentum cancerous satisfies the continuity equation for continuity equations 1 for each type of stuff will talk about OK will come back pressure while is
it said Friday that this is very low and the firm's probable and urging movement was No. in the 1st 6 days got that get hit this new here does not tell us what quantity Rico about we know we talk about electric charge so we could write this another way we could say this is jay charge New OK I just particular Peotone remind us what stuff we're talking about the new over here tells us about the direction of flow could wants time like density of space like this for all of the 2nd indexes thing which distinguishes full along with taxes the 1st index here tells us what quantity we're talking about saying here see the art houses were talking about energy this and trousers were talking about the flow of it along the axis and how much it was falling out window Oriental but flight stood to remember John to Europe and North America to secure their equal
that symmetry we don't have to remember which it's easy enough to remember that 1 of these must be 1 of these must correspond to the fact that its energy and the other does flow but not important to remember which is which because they're interchangeable to this OK so now let's put returned a bad said that if I had not known that the entire images of the entire matrix Turks's the correct will come back to its structure of who want maybe North America but a grant and some of the meaning of its elements but for the moment what would let it now Is it the notion of energy density is some incomplete list part of the multiplicative things part of cancer of course as part of the cancer of the right-hand side of this equation was part of the century the left-hand side must also be part of it but when illegal haven't cancer equation you can't cancer equation that says some particular component of a tense is equal to some are there the same component of but that the bad is an example let's take some particular component Our for vector a USA 83 it's a former rector and this is the 3rd component and I a the laws of physics that says that a is equal to be but the 3 being commanders E direction does not make sense as a law of physics well it only makes sense as a law of physics if it is also true that 82 with people's be tool and 81 because we want what is that why can't you just have a lot of it is the 3rd
component of vector along the z-axis is equal to the 3rd component of some other factors they must have the other 2 components are equally cancer simple that if if it is always true away in every frame of reference that the 3rd component of any is equal to the 3rd component of which Truitt every frame of reference then by rotating the frame of reference we can rotate 83 that we can look papers the 3rd axis until it becomes the 2nd axis of its joint every frame of reference that 83 is equal to be 3 then a tool must be equaled beat too and they want must be equal to be won if it's to be true in every frame of reference better example of white equations need to be Cancer equations of the form a sober and Eagles beast of em all our vector equations full vector court OK when you go to relativity the same thing is true it even concluding that kind component of a equator if a what show for example in some frame of reference not in every frame of reference that a certain form back there I was a former rector this is the time component of it is always equal to be 0 it every frame of reference and the only way that can be true is if all components are equal in every frame of reference a new physical to be new for the same reason Lorenzo transformations are not so different from location to our and I'm less all the components are equal you'll always be able to find a frame of reference in which a North will not be equal to be nor unless all 4 equal soap good laws of physics must be canceled laws of physics Our particularly if it to be true in every frame of reference now healing having equation that involves a right-hand side which is a particular component of some cancer it's a component Nought not component of the year the energy momentum tensor let's not worry talk much about whether his left side was just a little just a guess of what the left hand side might look like but the right hand side is the energy density and what you expect to be on the right hand side of the equation but who owns a sorry override so equations for the right hand side of Einstein's equations must involve not a particular component of a cancer but was generalized to something that involves all components of the cancer so that means Einstein's generalization of new must read something like this hope don't the right hand side must be G say they pie G times team you know a special case being when you and who are both equal to kind and then that becomes the energy density but the equation to be true it every frame it has to be considered equation what has to be on the left-hand side of what has to be on the left-hand side must also be cancer with 2 components rank 2 otherwise the equation does make sense on the left side must be some cancer with the same kind of cancer structure must be symmetric because the right hand side so that they can have whatever properties right side has as but it's not something which is made up Out of matter it's made up out of the Matrix something which is made up of the metric it has to do with geometry and not with masses sources so the left hand side we will just say will call capital Jean the only thing we know about it is made up of the Matrix it probably has to derivatives in it to compare with this year but it will involve the metric in some form and very likely to derivatives of file that's the kind of thing we would like to to find and put on the left-hand side I would find a good candidate for a week and then eh and they have In a situation where a nonrelativistic physics should be a good approximation does is gene you know reduce to just Dulce pledging on perhaps bars clapped but it doesn't than we throw away in China find a different role so we need to know what kind of thing G. New New could select starts next we possibilities Jamil known as a cancer made up out power however the metric it has to derivatives where Mr. must have some terms which have to derivatives to make it look like that so much as by itself has with 2 derivatives what kind of cancer going make ever met and 2 derivatives products or a river rate talk about 1 it was the curvature tensor may remind you about the curvature of the curvature tensor was made up of the Qwest awful symbols for our purposes tonight I just writing down these equations as
reminders body histolytica staff all symbols and not just remind you what they look like the physical toll 1 half GE Sigma Delta this deep cloudy town means debug BX towel our jails class derivative with respect to new ways of the Jedi Delta towel the 1st 2 turns a remain the changing how and when the last 1 was derivative of of with respect 2 Delta doorways biology tell that would be the only important thing right now this involves the derivative of a G involves the 1st derivative of G 1 derivative next what about the curvature tensor underwrite the curvature tensor but they all important part of it is of involves another derivative so here it is the curvature tensor Arnold glory are you know now and this is nothing which tells us that the real curvature if any component of this is nonzero anywhere in a region of space the spaces curved or will calm waters tickled to its equal to think Dean Newell jam and Soon the Miners these new glamorous said new tell and there was a mother gallon said de Lemieux tell land don't a much Jose reason regretting his dairy farmers is just a general overall structure which is interesting for reasons Lambert said Demirel new town where some 80 convention a assault maimed pointers the staff symbol which is not considered Her 1st the derivatives of the rhetoric and the curvature tensor at 1st derivatives of okra offal and also has things which are quadratic in Acra staff or symbol to pray this means these parents Europe's 2nd derivatives of met a derivatives of the river this Fahrenheit has also to derivatives squares of 1st derivatives this in the kind of thing would like to see we like to see and are we like to see cancer which involves 2 derivatives emetic connected to a derivative of components of America that's a candidate for components of this various components of our candidates to appear on the left-hand side of this equation but late ot has 4 components the curvature tensor 3 three-month curvature tensor has 4 components the left-hand side of this equation only as allowed that components why did is right hand side compartments so this c and b the left-hand side of the equation itself what can you do it to make a thing with took components so you could make a thing with 2 components by contraction a contraction of compulsory remember role is used that stigma for example about a play or have you could set singly equaled the Powell and some you get 0 if you actually plug-in what we curvature tensor is if you Cigna equaled the and some you get 0 if said similar to know and you won't get stigma you give 0 you get something we can call the cancer are new house the uterus called track mistresses you kiss is it all exactly was coming next week bill the cancer from a cancer with 4 components you could build a cancer widow components by contracting indices but they have that be careful that you don't give 0 you will get 0 some of the signatories of the of various minus signs that appear here In fact What's the various minus signs that appear here will warning that giving you 0 when you could track signal with how they will not give you 0 new contracts signal a new war or a signal of new but the to cancers you get by contracting signal with new and knew what happened be the same cancer apart from
a sign was really only 1 thing you could build a pyramids of well-known most called his only one-tenth that you could build to derivatives acting on the metric which has only 2 words To win this is called the Ricci tensor a contraction of the three-month cancer has less information every month and there's a lot more components in the Ricci tensor there's less information leading to say it doesn't itself commuters Zero without before being 0 Oh this is called the Ricci tensor end those always if you have a cancer you can erase and Lauritz indices that means is also a thing called is also thing called are new also are you tower with both of them raised Yukon laser Laura indices using the electric considered themselves US US downtown for each such was a mobile 1 with operate indices another factor but the Ricci tensor is ever happens to be symmetric in particular are Newtown vehicles are tower now they just checked by by by using its definition I don't know and I don't know a simple quick argument about it follows these things through complicated by this Ricci tensor is symmetric a property that has fine the way it is and so it is possible left him sorry it would be the Ricci tensor question mark a pie G team you now there's another 1 that you could make his another warned you could make a motel so the 2 could make it was not unique far only 1 of only possibility and that's to begin by contracting muon Tao Lori if you could attract new and towels by multiplying us the subject is called off its scalar it's a scalar that's are you knew which is also equals GE New now turned all are whoops town of army towel Jean Utah the action of the Jedi lowers the index Powell and mix interview In becomes just exactly this thing here is of the same object this is called the curvature scalar scalar and has no indices left at all so it's not what we want on the left-hand side but we can multiply bite Jean Uno multiplying it by Jean you does give us a cancer that's another possibility not recommending either of these at the moment I'm just saying from what we set up till now either of these could be possible loss of gravitation both involved 2nd derivatives of the metric tensor equaling something on the right hand side which looks like a density of them energy and momentum OK which once a week text while we know 1 more thing we no 1 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 a property that only disappears foes room walls of systems Meridith as a flower then we're forced to that conclusion that Dean you call area derivative of tea you know physical 0 continuity equation new team you was the continuity equation from which it follows that he knew Genia Miller is equal to 0 so the 1st thing we could do is we could check whether either of these 2 cancers satisfies this relationship if not In the left-hand side simply can't be the right hand side on lest we give up conservation local continuity of the MGM moment buyout checkers form for you but just check this 1 and I'll tell you what the other 1 was let's check
what's calculate being you lot of GE new Middle Ott it's opposed parentheses around OK 1st act call varied derivatives satisfy the usual product law that this is equal to Dean genial New Times are lost genius these new law just product rules for derivatives called variant derivatives of no exception this is now what about the coal the river of the metric tensor the of the cold variant derivative of the metric 0 0 that's all we started we calculated the other Grosof will symbols are starting with the assumption that the call their derivative Jews 0 and the reason for that is because called variant derivatives are by definition cancers which means the special good frame of reference all are equal toward ordinary derivatives but the ordinary derivative of Jedi in the good frame of reference of good frame of reference being the frame of reference which the derivatives of GEO 0 so the call G is equal to 0 this term not there now offers a scalar the called variant derivative of scalar is just ordinary derivatives scalar don't have any indices derivatives covariant derivatives of scalar so just ordinary derivatives were right here are plus the right away Fiat about this this is just tickled the GUE times the derivative of all was certainly in general the derivatives Of the curvature is not 0 in general the derivative of curvature of curvature scalar curvature scalar was constant every we know that's not true we know that they can be geometries which are more curve in 1 place less courage other places even flap in some places certainly is not the case that derivative aura is identically equal to 0 and the gene here doesn't help you come less Yukon lawyers commode Virginia New get rid of it I would just say the derivative of article was 0 is a refined 1st of all the call variant derivative called variant derivative of this guy over here is not equal to 0 but it is equal to just she knew Dean you are so does or no bad what about this work well we go the same things we calculate being you of are you who edits a little harder but not much of a little bit harder alter you what the answer is that each it's a quota of 1 half G new new game you are begin it can't be 0 for the same reason that this fund can be a happens the exactly 1 half variant derivative of the corresponding covariant derivative but now we know the answer everything Genito art and subtract off 1 half on you knew or better yet take on know who should attract off I where is to warn have formed by our case or ignore arm derivatives derivative new Fiat is equal to 1 half yard as they arrived at by by now have I now have a obvious thing to do we combine this with this and know With some coefficient genius and then we will get a thing whose firm appropriate derivative it is equal to 0 so where's Jimmy G is on 0 minus 1 half OR if it put death that's a theory that there is nothing else made up of 2 derivatives acting on the rhetoric that has the property that them it's called variably conserve this would recall coal variably conserve our just is nothing else that either so I OII pickaback cost you have twice it over half of it 17 times it but now it just becomes a question of matching this equation condones equations In the appropriate approximations where everything is moving number worker mystically man 1 of the possibilities if there is some correct numerical multiple of theirs which matches this or there is and if there isn't then we're in trouble the were trouble Salem matches if we look at the prime time component of prime-time component of this equation has all the right hand side it has he won't knocked on the right hand side is role we take this equation in nonrelativistic limit everybody moving slowly not too strong gravitational field With plug in and we hope that with some appropriate numerical coefficients here this equation in this equation are the same in the appropriate tho it may be the answer is yes they are the same and the the same with coefficient war commerical coefficient 1 just a piece of luck but it turned out with coefficient Warren Buffett there nothing deep about that on this what Einstein this was Einstein's calculations he knew what was going on pretty much but he didn't quite know what the right equation emotional wars I believe in the beginning he actually did try our equal to team here and there will eventually Sharon realized that it could work I remember me weeks of work at the prompted the wall of this but In the end he discovered capital Jr. no calls on you know like this 1 have are misses call the Einstein tensor Susie Ricci cancer and this is the curvature scalar so this is now known as Einstein's field equations they don't reproduce and 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 can evolve Nevada involved energy flow can involve momentum density and can even involved momentum flow not as a momentum flow or the energy for are the energy flow through the momentum but even the momentum density are much smaller than the energy density while I say that it has to do with the speed of light formulas to
do this to put the speed of light into a formula was just like energy is always huge because he gets multiplied by C squared but on the other hand movement was typically not swallows huge figures is just mass times velocity so velocity is law if you're in the nonrelativistic situation velocities energy is big energy density is by far the biggest saying the other components of the energy momentum tensor are much smaller typically decreased by powers of the of the speed of light on the various non relativistic situation was the only thing that's important right inside a frame of reference where resources are moving slowly in a frame of reference for the sources of moving slowly the only important thing on the right-hand side His wrote it's also true that in the same limit the only important thing on the left-hand side is the 2nd derivative of Ge Ge nor so nonrelativistic limit these things match but outside of the nonrelativistic limit things places where sources are moving rapidly or even places where the sources made up of particles which are moving rapidly even know the whole thing maybe not moving so much other components of the energy momentum attempted to generate gravitation they do generate their curvature and not just energy which are rich or mass appear every sometimes court but he utility can stay in July that would be a little bit of fluid is Allegis is Alice's lady will visit this is structures because there is some is who would come with this the was a yet world without using cover versions villagers will want it appears Wilbert different logic is the continuity equation basically comes from the idea of conservation and local conservation none of his crap where you have something disappear and reappear on her office that logic is just as good and relativity as it is in the nonrelativistic physics staff having to pass out the resort disappear disappeared showed some sense that continuity equation is more fundamental from this point of view but there another point of view which is I which will talk about maybe next time which is B action formulation which is much more beautiful and amateur rock bands where we introduce a principal least action for the gravitational field where or of this just pops up hard calculations easy but there a bird popped out they argued that the curvature scalar that's all just curvature scalar was Granzien density Kurt the war the car said the curvature scalar plus some things from matter from the matter for from sources that the case full of course that's a Kassar yesterday as bad it says they are all very it's not a case that is also not a mathematical hacked but is visible things well I can't think of any simple physical argument for error but it is a fact of life said 5 0 I know it's it's it's an easy factor only the only concern this was not from remind of the cancer and build involves only to derivatives of the year metric always three-month curvature cancer and things that descend from it by year by year contraction so yes Einstein Newell there had to be built about the curvature and it's not hard to go through to exhaust all possible things that you could do with a curvature tensor essentially did it all possible things you could do with the curvature tensor to make a a thing with only 2 indices with only 2 things you can do 1 of them was on the other end this not hard to prove it a very straightforward so Einstein presumably know that that this particular current combinations satisfy the continuity equation that was Einstein so we must either do a little bit of work the calculates this thing over here a new army and what he had to do a little bit of work to to calculate that I assume lawyer did was just plug and would take about 15 or 20 minutes To do it as usual the principles a simple but those by the time you manipulator early in the season Namur Christoval symbols are worried about getting the signs right so what built take give good 15 minutes to get a start so I can envision him having done calculation noticed that but agreed with this calculation and that if he combined them the right proportions you have ordered needs there said there is another argument the other argument a much more elegant but that this was the 1st round of things that he did not say what is needed a cash has for disease says he had made a full yes 0 yes 0 yes 0 yes and here by another weighty out right OK so we'll Guillereault world who composed this is quite a bit of content or arms but let me point out 1 thing that just as In Maxwell's equations there are solutions which don't involve sources you can have you could find solutions of Maxwell's equations that don't have any armed or you can find solutions in regions of space in which there are no war no sources let's consider the case with either no to war will ruin a region of space where there are no sources outside of all sources then on the right hand side we have 0 so we just show you that there's a simplification in that case is a little simplification but equation does become simpler in that case let's suppose that team you know is equal to 0 another words are minus 1 here G minerals ah is equal to 0 or better yet aren't you know toward him Jimmy and are now let's calculate barred by on the left-hand side contracting Newman equivalent of lowering in index here left the DeLorean underside also in this setting new wiggle you dear amyloid GE is with 1 upper and lower index it's a crime occurred Delta are go Bakhtiar notes you remember we GE with 1 Barbara and 1 Laura index at the which means chronic adult for now required to Delta in general relativity the Chronicle is always considered to be a thing with 1 1 exposed to send zeroing is not equal to no
1 equipped with the unit matrix the reason OK were the of Ottawa let's now contract you would know that means set new equal to and somber where Wagoner
left inside left Kansai they get are on the right hand side get 1 hey F are kinds is object Jean knew what they had Delta Nu meal was dealt a new for for therefore pieces if he leads Delta Delta Nought Nought is 1 of the 1 wonders 1 built the total was 1 of the 3 3 is 1 this is just 4 the latest doping equation Aura's equal to 2 are the only solution of that is that are was
equal to 0 not the curvature tensor now even the Ricci tensor but just curvature scalar vector was not implied that be Ricci cancer is 0 in only implies wouldn't wires which is the Ricci scalar is equal to 0 in which case in this special case when this is equal to 0 you could drop this 1 Evgeniou no R is 0 Einstein's field equations become 0 Lille simpler was easily called the vacuum case that back in case means in regions a space where there are no sources in regions a space where there are no sources be called Einstein's equations is just the equations that are knew was equal to 0 the solutions are marked revealed they contain gravitational waves with no sources dislike their electromagnetic waves but another example is the what chilled metric now the shortest electric itself is analogous roughly speaking analogous to what Tulip point man outside the point maps there is no matter nothing so just like doing equations lower be everywhere outside the mask the equation is the same as it would be for just empty space Schwartz still metric is also won which everywhere is outside the singularity v equation that satisfied is he is a vacuum Einstein equations or simple thing it's not simple to do with it a real nuisance to boasts a conceptually simple thing to do was to take the Schwartz children at earth and calculate arm you knew and check that it's equal to 0 you will find that if you take a short film rhetoric but too much Jeannie sit down and spend the rest of the day calculating Christoval symbols and then curvature cancers and contracting them or put it on Mathematica and you find out exactly equal to 0 Golf course ambiguous singularity singularity of it is so crazy the make sends all but there a components which a contracting become infinite they are so alike point to sing. were and undefined at the origin have a value but every away from the singularity yes the Schwartz shield rhetoric is what is called Ricci flat saying this is equal to 0 is sometimes called Ricci flat it is not the same as flak from gravitational waves satisfy this Schwartz metric satisfies it accepted the singularity Mendham that that's basic facts about the Einstein field equations that will quit the idea of gasses EDG does it use a mouse trap he said mutate OK but the idea could right OK a that a that OK until we get a lot earlier Nova doesn't matter words of as advanced as our no minus 1 AFG Nuno are is equal to 8 Chi G T T came now we're we're going to contract the contracted these 2 indices is is gonna give us all minors so are busy quota 8 Taipei G T New York 2nd track the index for for the the same works the same thing team you know Jamie mineral OK so that now tells us what CBS's minus all my yes aura was a pie G let's just call it T he is
by definition the scalar that you get when you print track the 2 indices here so a category of art is equal to minus a pie G T and now Wickham put that back into the Einstein equations put this backward foreign leaders you minus 1 half genial cars are which plus 8 GE T vehicles Thai G he knew no warm and now works take this thing here and shifted to the right this was a plus sign here so we get a G seeing you know a part G multipliers team minus 1 hey have Jean Nuuanu tea where teeth as got my contracting key so Oh but it is a we get another words Einstein field equations could be written with aren't you know on the left-hand side by former right-hand side you have to compensate by subtracting 1 Jr. Times take the yachts called a trace of the energy momentum tensor only when Burma sigh we are here trace of the engine moment of a case of with 0 it's 0 0 for radiation electromagnetic radiation 0 0 when the mass of video and when 0 for massless particles like a photons Gravitt is a true electromagnetic radiation it would be 0 2 0 the particles With may ask the energy momentum tensor this thing is not roh so this has brought about truck all G we've got to get rid that here are removed and has to do with Gordon around the world go currency held out of your control retention the owned I don't most important I don't any Sim particular physical significance geometric significant I'm sure there is but whatever it is I'm it's it's not really very transparent there are much simpler objects in the holy month and on the other hand the kind of difficult visualize their individual meaning so I no I don't know goods simple way and I had a flat address to add a playoff drought right strong U.S. whether you lose information well the answer is yes you can have geometries where on you know is equal to 0 where the three-month cancer not equal to 0 an example that will explore a little better it is our own gravitational waves gravitational waves just like electromagnetic waves they don't require resources request really we really expect the real world an electromagnetic waves made by an antenna or something but solutions Maxwell's equations you could just have electromagnetic waves there just propagate from infinity to infinity and just most of the same way you could have gravitational waves which also have no saucers those gravitational waves satisfier army in the week was 0 but they are most certainly not flat space is all sorts of distortions of space going on so I think maybe a possible somewhat satisfying thing to see what but geometry that has offered new equal to 0 0 much for which they curvature tensor itself was not equal to 0 and go explore little bit were gravitational waves looks like that way same thing of course is Jewish what rhetoric as long as you stay with the singularity is something that is real curvature Vera a tidal forces also stuff but the are musical 0 so yes on you know has less information than the curvature and Vasily depends on the dimensionality in 4 dimensions there is less information in these Ricci tensor the three-month answer because that 3 dimensions the amount of information as a saying you could write there are you could write 1 in terms of the other in away and true mentions all of the information His In scalar jurors is a scalar and you could make the other things out says In the southern Beirut by the rich and daughter of greatest but your competition that give back out of the house that he said that with all that refers because there's so disease may have had reached a the point where it
s cause left I of palace said a revision that it means it means that a given the source so they can be many many solutions many of which were they all have the the same left inside you know but they simply have different different physical properties for example the simplest situation is to what this is near 0 does doesn't mean there is no gravitational no interesting geometry or not allows gravitational waves roughly speaking any energy momentum tensor you put constructive solutions and then eh AT & gravitational waves on top of roughly speaking that's not exactly true but there are both roughly speaking troll that any solution you could always ended gravitational waves so the gravitational waves must be something that however contains more information than just Ricci tensor they do you foot for now but no car will now come on now the cosmological constant there can be thought of as part of a team you know so we have to rearm said anything about cosmological constant whether announced their Le yogurt from this point of view the cosmological constant can I be thought of as and energy momentum tensor components to energy momentum tensor With team you know cosmological for the cosmological constant would be some Lander and junior so then if was a cosmological constant you might write a party G times land their are friends G middle this number here is just a constant with just get rid of it quality somewhere I guess it is really the cosmological constant use you could shit it only depends on geometry number time geometry Ucon could shifted to the other side of the equation and think of it as part of the geometry geometric side of the equation for you coming on the right side and think of it as part of the energy momentum cosmological constant we fervently the way it 9 or is it an observer that um source said Shell a poor says sure we see friend its probably Americans papers all which I have homer multiple but don't was done to your see that done it was done by a he didn't have the Schwartz solution but he did have the approximation of the Schwartz solution at a fairly large distant nervous they the surface of the sun is way way out of a distant much larger which what you radius of the sound of course son is not a black hole but outside outside the solar radius the geometry is exactly the same as the Schweiger tree he didn't have the short DRG but he had a pretty good approximation to it that the or 100 times larger distance what radius of more private has a million times larger than which were children so he knew how to make a good approximation that was truly far from these far from the center of a gravitating object end then he just solve audits and that they have geometry again and again approximation anything that allows you to do that is the fact that the signers it is so much bigger than Schweiger and that means the corrections from Newton are small figures just small correction from Newton and the work our in a kind of firmer perturbation theory perturbation theory me just more corrections to something you already know most likely what it did was just take a capillary in orbit for a light rain just have risen in straight line and do a bit of perturbation theory Mr. Mercurio of the bending of life at which Mercury Yasuda darned out just took the kept Larionov ordinary kept Lorraine or bet 5 the Newtonian solutions Class a small correction and fit the small correction on the left side of the small corrections but then schwa immediately think within a year of U.S calculated his exact solution of the equations from May you could do a much better than OK you know pretty much theorem they 1 OR squared potential what's more only power for which allows the audits to be closed without a procession of the year of your ripped curious fact some are accidental
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