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# General Relativity | Lecture 9

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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
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
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
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
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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Eisendraht
Vakuumphysik
Source <Elektronik>
Elektromagnetische Welle
Halo <Atmosphärische Optik>
EDG, Entsorgung Dortmund GmbH
Dotant
Verzerrung <Elektrotechnik>
Dürre
Singularität <Meteorologie>
Regler
Masse <Physik>
Magnetisches Dipolmoment
Kraftfahrzeugexport
Sonnenstrahlung
Fahrzeug
Besprechung/Interview
Erdefunkstelle
Photonik
Remanenz
Störgröße
Jacht
Leistungssteuerung
Teilchen
Woche
Prozessleittechnik
Empfangsantenne
Fuß <Maßeinheit>
Passfeder
Brechzahl
Starkregen
Kontraktion
Multiplizität
Schall
Kaltumformen
Umlaufbahn
Kraft-Wärme-Kopplung
Einstein-Feldgleichungen
Längenmessung
Elektronisches Bauelement
Rauschzahl
Gleiskette
Schiffsklassifikation
Videotechnik
Motor
Strahlung
Lastkraftwagen
Jahr
Newtonsche Flüssigkeit
Source <Elektronik>
Elektromagnetische Welle
Ersatzteil
Papier
Kapillarität