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# General Relativity | Lecture 10

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question is so as to make nite there more or less by accident there are exactly the ones that I want to address tonight weak gravitational fields Ranieri vs. nonlinearity and gravitational waves let's begin working out the equations of general relativity is all the ways on pleasant not going to do it on the blackboard Perry a word there are still a blackboard even for simple things than they probably would not be terribly illuminating to learn this subject really how the computer however solvent equations and so forth thing have to sit down and do it for the other hand the principles a straightforward enough and it's easy enough to say what you get when you do solve the equations so that we will talk about gravity waves our by writing down the equations and then writing down solution now we're interested in what could be called week gravitational waves weak gravitational waves means that amplitude of the gravitational waves are small enough that you could make approximation such actors the amplitude square of the gravitational field 0 or quantity is small and you're expanding equation about the smallness of our quantity the the usual rule is to ignore things which Ohio water In that small quantity but we start with an equilibrium what we're talking about fluctuations of perturbations about equilibrium situation and the simplest equilibrium solution aligned with the threat aren't equations 1st Einstein's equations but say the case without any matter Noah no energy momentum tensor and the equations of motion and just aren't you know minus 1 have Jim you know Our is equal to 0 Kobe Einstein cancer do you know of 0 they can be simplified if you take it traces of both sides are here is no trace of the year Ricci tensor I don't interest both sides you will discover that are the scalar curvature R is equal to 0 . you know that you could ignore the 2nd term in the equation equations said a doesn't really matter because we're not gonna write down the details anyway but this is the equation of Einstein in a context in which there is no energy momentum tensor on the right-hand side by Watson equilibrium situation and equilibrium situation 1st of all means a solution which has no time-dependent is really are and which also doesn't have any matter on the right right-hand side matters another word for the energy momentum tensor right-hand side not right hand sorry there's really only 1 equilibrium situation equilibrium situation is just empty space empty spaces tiny independent BTR and these entities thereby empty space I mean in B flat space no curvature no interesting gravitational field and that case the metric g and now I'm not going to say the metric is equal to such as such the method depends on the quarter to you if I wrote that the metric on a flat plane is just a crocodile assembled you would correct in saying no the metric color flat plane is not be Crocker Delta a possible metric expressing the flatness of space would be the crocodile the values the coordinates the that that would not be the crime use Kurt coordinates used polar coordinates used any other kind of course that's what's special about flat space is that you can find coordinates in which metric customizable form the same is true in general relativity flat space time with no gravitational field there is a choice of coordinates in which the metric has a simple forms of aid and you know just remind you what a them you Louis 8 immune Nova is a matrix that's a fright matrix laughter I think with notations we've been using in this class I forget some
1 minus 1 minus 1 minus 1 0 0 0 no sorry correct 1 0 0 0 0 0 1 9 1 0 0 0 0 1 0 0 0 0 1 which throwing column corresponds time the first one T X Y
Z T X Y Z so forth OK so that's metric of flat space-time 1 might 1 and 3 pluses Anderson equilibrium solution it is a solution of the Einstein field equations the Einstein field equations components of curvature are equal 0 this is a geometry of the symmetric has no curvature of all so it is a solution 4 billion you would find a solution factor will be trivial step now let's think about something which is close to empty space far away some complicated thing is happening far away so far only to be interesting to us at least to be close enough for us to directly observed there's something going on AT binary Paul Soros a rotating around doing some complicated emission of gravitational waves closer by the gravitational field may be very strong even the gravitational field waves might be rather strong but you go far enough away the gravitational radiation the gravitational waves produced by this are going to be very weak was weak means week means that the trough metric can't be chosen again emphasize can be be chosen people a them you know class something small small means that its components are much smaller than the ones which appear here and that small thing is usually called each as far as as I know it's core age because each is the letter after G don't know any of a reason for 4 h h mu Nova on like a them you know is in general a function of position it's also what I say position 19th position and time at a function of cordon that varies from place to place their might describe away will come back to raise moment but wary doing this take this metric calculate from it are new new and set equal to 0 and that's give us an equation for now the equator and that you actually get from H H it's not big enough to fill a hole blackboard big enough to move to quite the unpleasant and some scanning I'm going to be schematic show you what goes into 1st of all waters off our ears really is a combination of components Of the curvature and not right to curvature tensor underscore remind you would contain the curvature tensor itself with 4 indices contains 1 turned which is a Durant is with respect to position with respect to acts on on a right which X it contains lots of different derivatives depending on the index structure derivative of the off symbol solar contains a 1st derivative of the kerfuffle symbol and secondly it contains the Christoval Sonal quite radically squared opera star possible times Nova Christoval so this is really offices as much detail as we do I could put picked in the indices inherent remind you with the indices go in there are various turns of took like this there are various terms that but as a good enough for us what about gamma what about the staff or symbol of you remember that your 1st contains a factor 1 hailed it contains metric cancer with component upstairs the that with component upstairs is simply be in matrix of the metric tensor itself 8 can also be expanded powers of each of the 1st contribution to it it is just in Istanbul itself so the inverse aid is inverse matrix and so it begins with G the minus 1 on be more specific America and never has derivatives of GE it has various kinds of derivatives of GE Malcolm G in reverse that contains again things like leader probably minus H but whatever it is it contains he ate a symbol plus a small correction be metric tensor with upper indices if you only have
Ada would basically be the same data and then age would appear as a correction 5 Sergey Mercier this would contain things like this is a legally equal sign which means not equal it would contain things like Ada minus a small Correctional plus a small correction report for us but then the of GE that only contains a because the derivative of this metric 0 contains a zeros in 1 derivative
zeroes in 1 0 themselves and the approximation working in the derivative of G just contains derivatives of age which call derivative of H OK let's look at what we have here this the Chris symbol it contains 1 turned which has won power of H incidentally H is small and will also assume the derivative is also small if you have a small number of small function in a differentiated that's equally small general has very shocked point or something which will show the doesn't so the theory of H it is also considered to be a small thing they had times H is wants a small beach the derivative 8 is twice a small but quadratic In the fluctuation are in the yard in a small gravitational field so ignore it too small to be important if beaches . 0 1 and derivative of 8 . 0 0 1 then aged times H is . 0 0 0 0 1 whoever is very small but we can ignore this a net h will just call derivative of age is called so the crisp Opel symbol itself it is just in the approximation of small grabbed a weak gravitational radiation the Crist awful symbol itself is proportional to some derivative box and collection of derivatives of H what about the Ricci tensor then the Ricci tensor well contained curriculum gamma
Delta mn website Annan instead effects wife on Elian there's nothing to prevent me from introducing court coordinates with little wiggle in well we could start with coordinates what's we start with coordinates which other good Cartesian coordinates let's call them X XMY will occur with contracts that's a lot more than they would introduce no coordinates next time and why Prime where X prime is just X a small change quickly call that plus f sub x of excellent y as according to change according change X prime is some function of X and Y and Y Prime some other function of X and Y at . change here at the change is taken to be almost no coordinate change at all plus a small Will correction the same thing for y prime musical Hawaii plus let's call best why aleksandr White I have not seen the space it anyway all I've done is change coordinates and perhaps put some little wiggle sofa 1st test of wills and then the no coordinate axes might have revenue Courtney lines might have some wheels and but the 4 if I rewrite the metric for the current components are of let's say you are let's OK let's take the the case looks assumed the metric in the Klein Court trips is just be X Times squared plus BY scores I'm working backward and supposing that prime coordinates organized coordinates the mice Cartesian coordinates and the X and Y coordinates Of the year of the 1st slightly curved form but was the metric the metric this is just a blackboard now outgoing space time just a blackboard the magic of the black His DX prime squared forced to wipe right square let's look out for fun in exon wife OK so what is the former work at work DX primaries DX front commerce DX is equal to DX plus the derivative of f sub X with respect to its call x sub x Kobe the X or Y funds DX and and likewise for White we figure out work DY Prime Minister DYE plus a small correction is year I want find out you find out that the X Prize square plus 2 why Prime squared is not D X squared plus Y squared it contains DX square post y squared but also contains order of VAX times this Andy times that if you work it out but you find out that this is equal to D XTY sorry the BX queer Post e Weiss word assuming that a small we're going to assume that the small and that means when I Square D X prime squared I Icahn drop the F X squares likewise for why don't wanna get is a correction and the correction could be called a each and VXM excellent another words just by squaring up PX foreign square BY prime square substituting in it you'll discover that is a small correction to the metric tensor now there's a small correction the metric cancer mean that the geometry of a blackboard change now it just means that beyond that the accord the views of totals in Ukiah worked out it's it's it's nice to work out little exercise worked out with the correction in that is dropping it anything which is quadratic in an dropping anything which is higher or not want to get you get that H & & And now just run over X and Y is equal could be be F M D X In Class D In my DX and that will mean nothing to you want to you try the work as an example onto you tried approve it this is the a correction for the rhetoric it's correct from Electric which is small white because of because I chose F to be small but it has the form wars a small perturbation on that but doesn't represent anything it represents just a trivial love change so there are some let's call them perturbations perturbations small change perturbations the original form of the metric which are trivial they don't represent likewise here there are small fluctuations that you could write down which just represent curvy linear cordoned space how do you get rid of that I get rid it and eliminate these the phony solutions which automatically solve the equation because they just flat space but which don't represent any real physics Eddie Dillard by imposing more equation more equations on the metric that so cut away and divide I would eliminate b b unwanted spurious solution I I could write down the equations it's not important just equations which tell you get rid of the spurious accord that there were coordinate ambiguities and you could do that a wide variety of ways wide variety of conditions that will allow you to eliminate on physical mean lists solutions of Einstein's equation once you've done that and you've imposed all of the Einstein equations once you've done that incidentally the equations become pretty simple and in fact Ole become as wave equations they become wave equations Camara several different equipped to become 1st of all with ordinary way just perfectly ordinary wave equations for the components of H. Singh continuity equations I wrote Barrow USAir race there be 2nd foray by DP squared minus D 2nd 5 90 x squared blah blah blah why Square D squared is equal to 0 facts of this summer let me right now In the correct form instead of fighting we have each year each meal each component of the metric tensor a rich component of the fluctuation satisfies a wave equation that means that all the components of the wave equation sir all components of the metric just moved down the actors Roark warrior always there are also some constraints constraints calmed the lot more equations from the
equations are their arms world there are more equations then 1 for each new until the reason that there are more equations is because you have to divide How wrote this spurious they solutions there because of the court of ambiguity want to that you find out that be physical solutions the ones that really have meaning classify them out for you supposing we have a way of moving down z-axis but suppose there moving down z-axis was a way of moving down z-axis look like wave moving down the z-axis Italy just a simple way you would just be fired is eagle something like some number times sigh of Katie Current X is a stark terms in team but came out it is in Manistee tonight as it is now said is a wave that fixed incident of time is just a sine wave and most family taxes unit velocity unit velocity here means a speed of light now each 1 of the components will have a solution like that each component will be proportional to sign Kate times what was Katie insolently after is close wave number but sister frequency of away is the frequency of away it's also the wave number the number of oscillations per unit weight the because could really think beginning number at short wavelengths of large a long wavelengths of more of a small pockets of that that's a typical solution of a wave equation and the solution of this equation here are all the same type H not is equal to some function coefficient elliptical coefficient trying not fight the number I each component as a function of teens E which is proportional saying things sank 8 times minus times and number which we can call each lot of new and here new window this is not a function of position it's just a numerical coefficient and multiplies the sign of human misery and Arsenal right that this is the nature of a gravitational waves each component of the metric behaving like a wave moving down the axis however as I said there are more equations additional equations is a good number of equations here and when you colors all you find out all before we do this looks just remind us something about the electromagnet magnetism and electromagnetism we do the same thing we solve Maxwell's field equations with song Max was a field equations for the vector potential of electric and magnetic fields but we write down all the equation we find some constraints constraints are called transverse our view of the fileserver nor transit salary of an electromagnetic field things it means of electromagnetic field electric and magnetic fields always point in the direction perpendicular to emotional not only are the solutions waves but that transverse waves meaning to say that the electric field and the magnetic field magnetic field . 1 way the electric fuel policy of golfing goes down the axis oscillating is a goes down the axis very similar things happen here the way you have to be transferred to say the way a transverse it means that the kind of components and the span Z components of H 0 0 only components of a which are allowed to be non-zero other components in the plane perpendicular the direction of the way so his substitute this year but all the equations the full set equations here including the ones which removed the fake spurious flocculation you find that a gravitational wave has a very very simple form does have this form a set of numbers fictional but the only component several hours this Izzy Texan Y I'll call jail eye injury represent the directions in the plane perpendicular to the motion of the way of the only components which are large part each I J climbs sign KM Casey Kizzie miners Kate t but but so if you're looking down z-axis if you looking down the z-axis you see in front of you the X-Y plane received from the X-Y plane as the x plane and the x plane but each ZTE lures a slice of slice by slice of a given location at a given time at a given location and a given time to the left of the two-dimensional plane and I slice the wave in two-dimensional planes as a good axis here at fixed instant of kind and what I have here is a metric for each Zee which looks like Where is it right over here it With only comport with the only components the components only in the plane perpendicular to the motion of the year of the year of the way and the composer simply numbers and each ZCU the the metric simply a set of numbers there's 1 more equation 1 equation that comes from Einstein's field equations and it says that the traces of H I J is equal to 0 in other words H X X plus each y y is equal to 0 that's it that's a whole set of equations and what it tells you is that the metric the metric of a gravitational waves as age not not equal 0 age zeal Nora physical 0 H Z easy is equal to 0 in fact H anything with either at time component or a space oryzae component Oracle 2 0 the only components which are not 0 Laura age X X H Y Y H ex-wife HYX comedic cancerous symmetric cancerous research brick and so H X Y N HYX of the same and finally HXX force HYY is equal to 0 which means that each HYY is minus HXX HYY physical commodity sex but trisyllable force amenities that 1st of all of these very With
Z & very holy was eager to if the wave is going down z-axis than by definition variations along the c-axis the sole HXX is some number to begin with punch signed KX signed hazy minus being sworn h Norwalk YYA sign K times in my and likewise Suarez H. North ex-wife sigh case that OK so think of a series of layers let's let's look at it at a fixed instant of kind of fix since time for example currently equals 0 as we moved down the exodus OK is a perturbation On the Net at the Met cancer is a little bit different from just a flat space metric it oscillate as you go down Zia third-down z-axis end the components that oscillate other components that have to do with the metric of the plane perpendicular to the wait what does it mean the have HXX well with just a case of accepts differs MHX acts that means the component of the metric is a little bit different than 1 the XX component of the metric was originally warned that Ada Ada XX it's 1 plus this little bit Obama bus so big that means that proper distance along the X axis are a little bit different that actual distances but say between x equals 0 0 and 1 Sea between X equals minus 1 and 1 the actual disk and the actual distance remember the metric tensor is thing which tells you the actual distance between points the coordinates the labels the actual distance between point minus 1 . plus 1 is a little bit different then it would be if this perturbation Syria a little bit larger separation or a little bit smaller separation depending on the sign of age except for example HXX is positive and the actual physical distance between here and here would be a little bit larger than 1 meter supposing working in meters that would mean the meter doesn't quite extend from minus one-to-one What about GYY well with GM will take HXX to be positive if HXX positive a yard to get that right the distance between minus 1 1 was a little bit bigger than a meter and therefore a real meterstick would look when without look it would be a little bit shorter the interval between minus 1 1 not the meter stickers necessarily shorter but because the coordinates of such that bet on that surface the distance between minus 1 plus 1 would be a little bit bigger than however GYY GYY is equal to 1 minus H X X minus HXX because Einstein's equations tell us that some of these 2 are equal to 0 that turned tells us that meterstick oriented along the Y axis what's safe from white equals 1 2 minus 1 here's again warned minus 1 meter stick would look a little longer the distance between plus 1 of my 1 would be a little bit less if you only have the 1 he and so are meterstick over here would look a little bit longer if it now this would be this will be nothing but a squeezing of the coordinates so that you would be choosing coordinates along the x-axis with meterstick was a little bit less than 1 unit and the music along the Y axis was the big greater than 1 will be nothing at all will be Nolan nothing significant accept the the fact that it changes as you go down the axis as you go down the exodus from 1 point or another or if the wave passing you imagine that even yet the wave was passed in the wake of his passing these sticks right after a brief amount of time these beaches would sign that sign of assign wave oscillate so after a brief interval of time 1 half cycle of the ways 8 X X has changed sign HYY his change sorry and the result is after a brief interval of time meterstick will look a little bit short in these coordinates along the y-axis a will belong along the x-axis yacht readers it's not know what express that little test In fact a little test ministers would go back and forth but not because the meter stake is Elliott 1 meter long always 1 meter long but right but an oscillating leave is going back and forth perturbing real waiver real wave of curvature this is real curvature and if you like Mr. Guo Li told Maginot around well the time dependence of this way you've really does exert tidal forces that tidal forces a wave has curvature of the waves moving past here that kind of tired force in the nature of the tidal force is actually cause meterstick be compressed stretched compressed stretched the way that compressed and stretch is that when it's compressed horizontally stretch vertically when it's compressed vertically it stretch horizontally so if you like this wave going by means of all art for that matter through which is to test masses really were just 2 test matches the 2 test massive oscillate back-and-forth here from it instead of taking a meterstick Turk a square piece of plywood square piece of plywood would be deformed this white and red river this way and then reversed now it's interesting there is another solution was another solution where it In which but not H X YEAR nonzero but in which ex-wife but is nonzero that would be a metric which would look alike crocodile filter and then what they'll IJT a little matrix which haven't HXY over here HXY over here still has HXX 6 y y equal to 0 knowledge XXH for 1 if you think about a little bit just spend a little bit of time figuring out what this means it's also the exactly the same kind of thing except that we accept that the squeezing stretching our
if I knew the distance the the actual proper distance associated with the X and the actual proper distance associated with be while I was a proper distance associated with them X VXC but square root GXX the square root of GXX DX period that but make it bigger Mr. the square root of GX X DX and this distances square root GYY wife cell what's the area of a rectangle square root GXX squared you DXT why it's not just CXT wife important thing is it's not just the exit to want no more so than the distance between 2 points is just XOD while the actual physical area is a square in this case is a square GXX GYY now I use the case here where it was only GXX Angie why there could be other components of the if you work it out this is a special case this is a special case with no off Bagwill GXY terms of strips of this way metric matrix GXX GYY now there may be some other components year GXY GXY thought I tell you it is that when there's no GXX and you want to share with was not off diagonal components the volume element involves a product of G X X GYY you guess with the general formula for replaces the proper thing determined determined GXX GYY miners GXY square the determinant is exactly right so if you actually work out the volume element of a little square like this or rectangle BXT White they actually answers not GXX URY GXX GYY miners G X Y Square which is as the determinant determined of the 10 it's usually just written with Tom bars to represent determinant Gee that's a formula for the volume element of volume of our small differential be ex ante why what would be a B and this can be a curved space the 1st any space for ever will rectangle BXT in coordinate space area or a higher dimensions it would be the volume would just be the square of G square of the determinant of G terms DXT worry if I have some large regions and I want a total volume of it I integrate into growth I love square root of the determinant of metric tons DXT why that is equal the area of the entire region and close with some members What about the volume of space if we're talking about the volume of words three-dimensional space what exactly the same except you have these here exactly the same square root of the determinant of the metric is called the volume element usually called volume already if you have a fuller dimensional space-time it's useful that finest simple a similar Carter concept which just involved the 4th component called the space volume it it's an interesting quantity it's invariant same in every coordinate system as an important thing about it if you re cordon the ties region change coordinates this quantity stays the same volume in some sense the physical volume of a region of space time of physical four-dimensional volume of regional space and this and the goal is invariant quantity on 0 invariants other quantities which are invariant them which will not change when you change coordinates are based on the squad if you take the volume element a new integrated with any scalar let's call scalar passive acts as aleksandr Cheney BXT Cheney Miss also is invariant quantity which doesn't depend on the court it's just the stuff you take each little volume element you take his volume and you multiply by the square root of starting the scalar field that the volume of each element multiply it by the scalar quantity in there and get them all out that's a quantity which is really not dependent in any way on the quarter to use even writing it down the uses of court action is a thing which is supposed to be in area action in relativity theory is always supposed to be the same in every quarter NetFrame otherwise the laws of physics which came out of it would depend on the coordinate system if the laws of physics was supposed to be the same in every quarter frame 1 way of ensuring that is to make action In invariant the quantity which doesn't change from cordoned the the court reporter 1st quarter so it's natural and to take the action could beef we call it before exiting some kind of squared and Janey the square of the determined from metric some scalar that was just take b Einstein's theory without any energy momentum and on the right hand side just pure gravitational field nothing else what kind of scalar cut I put their where the many dealers are there that I could make up only out of metric well indeed so that's called are the Safina exceed that's a scalar Brazil sir there was 1 more thing the other thing His any number 7 4 or 16 or whatever number you light so you could enter this numerical numbers numerical numbers are always regarded as invariants everybody 7 sees the same number and the number you can't hear depends on the laws of physics it is a law physics in itself was a law of physics has a name was called up they would know they were symbol is Simone slander now what's a cold the cosmological constant gets it termed the action which is just proportional to the space time volume itself turned the action when the Kansas the reserve is based on chair of the cosmological constant has not been and not included in these equations it is a solution is the equations without cosmological constant or about cosmological constant This is the action and Einstein's equations are completely equivalent saying minimize their make it stationary maggots
barred part of it is that very very few people know very much about Riemannian geometry I still know he had the he basically had to invent the whole of Riemannian geometry I think 1 of his friends helped them with food Newsom mathematics was Russell Grossman a murderer but but take years to put all this together doesn't sound like a lot of them are you he literally had to build all the ideas of curvature are so it took a long time here think I think he knew it was doing pretty well but there you flow back and look yourself papers are in English there were translate was you could find Boca Our Antigua finish work or you're doing rushed for more
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Gasturbine
Einbandmaterial
Vorlesung/Konferenz
Speckle-Interferometrie
Kontraktion
Flugbahn
Stunde
Strahlungsenergie
Pager
Kaltumformen
Waffentechnik
Elektronisches Bauelement
Tag
Rootsgebläse
Druckkraft
Gleichstrom
Flüssiger Brennstoff
Vakuumphysik
Jahr
Material
Magnetspule
Papier
Schall
Kaltumformen
Jahr
Ersatzteil
Textilveredelung
Papier