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The Einstein Equations and Gravitational Radiation

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around the the 2nd 2 and the typical thank you very much to the organizers to for these unlovely conference and the invitation so I'm gonna talk what and the weather is not it's going to improve the efficiency of the exactly Due to the other the right as a motivation to come inside so I'm going to talk about Einstein equations and then introduced some geometry into our PCs and of course we all heard about the detection of gravitational waves so it was confirmed in February this year that it was detected last September and I like to make a connection to let's say geometric analysis what we know from the PD .period a you analyzing the Einstein equations for consumer gravitational radiation from the mathematical point of view so basically I'm gonna talk introduced 1st Well what do you mean by space-time with all the Einstein equations and so again this will be a point of view of P of these hyperbolic set of speed ease from the geometric viewpoint I'm gonna make a connection with so what do we mean by energy in G Op general relativity of so may be not a straightforward straightforward notion to think off from the beginning we have to think about it a little bit I will talk about what I will call Noland Finnerty so let me just very curious Stokley multi-link this for the moment so if you think of gravitational radiation so they are produced during the murder of compact binaries like black holes for instance so there's something happening here you have to bodies black holes spiraling in they lose energy into the gravitational field so they radiate wastes and these wastes will travel at the speed of light along so-called knowledge surfaces so you can think of this as a generalization of polite ,comma basically so this will go a long not Libor surfaces and let's say when you're far away led 20 goes to infinity so we can think of what what and where we are sitting in an experiment so this is knowledge and so we will look at moment that's where the you're sitting when doing experiments and we're looking backwards in time along the small hyper surfaces so we would like to understand from our point of view of analysis and unitary how does our space look up look like out here and what kind of information can you gain from what's happening in the past year for instance in such a source I'll explain his Livermore later on well of course gravitational waves also addressed something called the memory effect that's a permanent change of the space time by the gravitational waves and we also hope that this will be detected in the near future and detection is 1 thing we hope of course for the future that this will be a new tool to really learn about other parts of the universe where telescopes cannot see OK then addressed some of the results in the asymptotic flat case so that's where we look at the galaxies for instance were we think we are here the Galaxy creates curvature but far away the next galaxies so far away so we can assume that spacetime is flat far away another setting is the cosmological setting 1 look at the whole space time the whole history of the universe what can you say that well 1st of all voices space time so we look at the Einstein equations on the left side and I'm going to talk about 4 dimensions so 3 plus 1 3 special one-time June so we look on the left-hand side so we have basic the chill metric object if you like to to have the Ricci curvature of the metric or something for the metric scalar curvature of the coldest Union left inside the Einstein tests and right inside so we plug in the so-called energy momentum Tenzer so whenever you have electric fields president wore fluids or anything else that is not given by gravity on left side so you plug into the edge momentum Tenzer and have also to supply the corresponding equations of France's if you have electric electromagnetic fields you Koppel your Einstein with Maxwell equations and you get the coupled on the natural system so we're still waiting the Einstein equations for the metric like to create the time either locally or globally in time and well will see so it's possible now to arm view of FBI's equations we usually right then as well if you look at it this way it looks nice and Compaq but usually write them as a system of hyperbolic nominee Pt as well if you
ask cosmological cultures so how does cosmology come while I standing the 1st in 1915 had written down his equation without this cosmological but then he wanted to study the cosmology and she was actually looking for a static universe of the time he didn't believe in anything that's moving the universe has to be static and to counteract gravity he then plopped in this land the term which is giving you an expansion basically so he got a static universe by doing that but for all purposes now so we think since the 1998 observations that our universe is a track actually expanding at an accelerated rate and this is modeled the by OK
and often unable just look at the UN's and vacuum equations so if the right inside your team news 0 yes equations reduced to the Ricci Keeler of the false based on being 0 and if there's already a lot of interesting information or questions you can ask about just the answer vacuum equations themselves OK well 1st of all and in order to study anything that it has to do with real physics we would like to do on mathematically rigorous problem of course you have to study the Koshy problems so it all starts with the crucial problem 49 Einstein equations and as most of the audience is not working in jobs some experts as an exception so let me read you a little bit What does it mean them to set up the problem for the eyes equipped so it's a bit different are in various ways so an initial data said 1 study you are so we think 1st of all of three-dimensional 94 also can think of to equals 0 you get a real money and complete manifolds look at us and talk with flat for instance that you're interested in to get a complete Riemannian metric the inducement metric on the initial hyper surface the specifies metric to intensive care unit and good right them and if you have anything on the right inside presence all the matter fields you also specify of Call of course these equations now the Einstein equations Koppel into 2 sets of equations 1 is the constraint equations which the initial data has to satisfy and the other set of equations or the evolution equations the ones you then sold into the future and so the constraint equations it's still cannot just pick any initial data but it has fulfill this constraints and only then you can think about doing and and evolution problem and Benecol she development of such data is given as a globally hyperbolic space time of this sort verifying the Einstein equations and the embedding given these things maybe just 2 months a tiny was 0 so we think of some space like hyper surface H 0 for instance chaebol or K and this initial thereafter society by constraints and so far the time field introduced us so this case is going to be the 2nd phenomenal form with respect to the well ,comma let 1st talk about hasn't topically flats systems as adjusts morning made it a few minutes ago so if you look at what happens for instance within a galaxy or a cluster of galaxies you think the next object is so far away you think your space space times than the flat very far too are very far away so what does that mean so we think the asymptotic lie-flat initially enough in the sense that outside of a sufficiently large contracts said so the age without kissed differ more fit to the complement of polls following off 3 so and then they're cheaper are the initial will the injured Demetra chaebol or and Kate half of the Katie bar to I J and K 2 0 fast enough of course I will make this more precise what type of decay what type of US public the fact that so that the 1st say that well if you look at the history of durability and so the price decreases were put out there in 1959 it took while there were many many things happening in between math and physics but it took him until 1950 and 60 where he wanted and her and what garage actually really solve the well posters for the islands equations if you look at differences in 1999 have Eddington's experiment from confirming the bending of light at Citron have on the physics side of Chiara many of these big things that happened or the expansion of the universe found by Internet 1927 so always thinks so have many of these successes and mathematically could ask Well what's been going on mathematically there were many people actually working on like shouted many people looking at the initial value problem but it's not that easy to figure out what should be be exactly the right way to put up the initial value problem so I'm skipping all this history but such interesting to read up on many people who contributed in many ways so basically 1952 that's the 1st rigorous general mathematical theory in terms of well-posted so imaginatively and then showed that if you take issue Boracay as such an initial dataset which is not well here this is formulated for the vacuum Einstein equations but it's been generalize too cold for most of the fields fields the right side so there exists a space-time satisfying yachts American equations With aging into and being a space like surface with inducement to Chibok 2nd environment for K and then the general form of what's so whenever you do work the approach of Soviets equations so basically you go back to 1 form of of the 1 of these welcomes this results of what is local results so that the other formulations given here social did you wanted to be out booked around in 1969 and came up with this formulation of the well OK so that's the 1st rigorous steps in in general would well let me say
something about energy and you are well most of us well and think about peace so often we need to find cost energy steps have to be controlled to get control of the solutions of of ours our equation that so if you think of doing something like that and you are and you do you take the most needy approach so things might fail so what's the problem here so 1 problem or interesting features that if I'm sitting at that point injecting G R I can actually transform away the gravitational field so it's just I can buy transformation I can make it 0 so how would you define energy at that point if you can actually transform away the field well that's 1 problem so you can think well maybe I don't look at that point 1st immediately integrate over his fear something "quotation mark local what turns out that's 1 thing we can do but also president publicly flat systems France's what's been understood quite well for a lot of interesting space times is you can integrate France's over space like slight slice or you can also look at what's still if you integrate locally over fear and get some massive energy of well expression and you can take the limit like the hockey mask arrived at them later and take the limits to Noland there's something else called the bond announced Wednesday will show up later and so there are mass and energy definitions for certain actually many interesting space times that we understand but let me give you may be a little bit of an idea of the confusion at the beginning by citing Einstein and 1 of his colleagues so I stand for related to some sort of energy were meant to be a America really actually for his for closed universe and so most of his colleagues at the time did not agree with it and here is what from what he says securities citation while the general relativity theory was approved of by most theoretical physicists and mathematicians almost all colleagues object to my formulation of the energy demand the not going to go into details but you wanted some kind of management and conservation for some closed universe but it turns out that's maybe not the right way to do it and so this is so bystanders shot from net but Mediasite no deduction shall frenetically Belgian mathematician and physicist who who derived from Johnson equations and sly 1st observations in 1927 that the universe is expanding it was not hobble hobbled paper 2 years later and he never talked about expanding universe OK
let's still stay a little bit more with energy and conservation so what what do we know so we can look at the calculated to do that we know from geometry and applied to the Romani encourage and well we take for instance twice constructed beyond the young Kennedy and get this equation on the left-hand side of the Einstein equations and that of course is implies by Johnson equations themselves and identity of this for so people at the beginning also thought Well is this energy energy conservation yes No does it make sense well it makes certain sense of some sort of some sort but to call this the energy and energy conservation is probably the wrong thing to do In Jr so maybe 1 more thing about that so thirst fearing if you think of it a little bit more generalized swinging well whatever looking for within some setting of a local motion theory we always look for on continuous group of a few more of transformations which leads the constant invariant right so to such an object we have been a quantity which is conserved an injury geographers 1st problem if you have the most general space-time you don't have any symmetry so what are you looking for you don't have any symmetries at all however nature is usually better than just the most general case for instance if you look at the nascent topically flat space your background Minkowski space has a lot of symmetries and you can start looking at maybe cause I I'm certain I summit recipient preserve basically is in a way or another in the manifold or controlled well here is maybe 1 more flight where Einstein Hilbert and was struggled over this energy component of the gravitational field so they were thinking about this very hardly at the very beginning and there's just 1 more "quotation mark by him and while this is in his book space time and matter of 1921 until looks at some of these components here so he says well nevertheless it seems to be physically meaning less to introduce this TI keys that's the artist instituted Tanja based on some specific local she and he has there as energy components of the gravitational field for these quantities are neither attends nor symmetric so in fact by choosing an appropriate coordinate system always TI case can be made to vanish at any given point also the differential relations referring to defer the virtues of Johnston's unit Tandja being 0 or without the physical meaning nevertheless by integrating them over an isolated system 1 gets invariant conserved quantities so you see there was a lot of thinking reflection about what they're right energy notion should be an already 1 suggestion is all well maybe we should integrate something over an isolated systems so that was already a good idea OK but let's not jump to more modern days well more than 61 and so if we think of 1 type of our notion of energy and momentum so this is the ADM energy linear and I'll momentum that is was introduced in 1961 on which these were Mizner so basically have this is the definition but the way you think of this you integrate these geometric objects basically atmosphere of overseer at led Oregon to infinity so you're in 1 space like slots and integrated basically over sphere infinity and you can define and energy when you're an adult momentum to in that way so you have to say of course this is this holds forum of many interesting topical flap systems and so energy and winning momentum hold actually very generally and for Parliament amid a little bit more if key to be defined so
why not Noland because I told you at the beginning heuristic was radiation like to understand all Finnerty and explain something about radiation here extracted from our space time so we would like to understand the course she problem and then see how the space-time dissolution of moment and knowing Mellenthin so but I can also fully MySpace time not only by splits a space like hyper surfaces like this I can also fully at MySpace time by not hyper services so an ideal draw them like a coat of course they have structured but you can think of them as generalized light ,comma for instance so this is maybe exude do want some of this light this knowledge for surfaces etc. that I can generate like that and again so 1 the gravitational waves travel at the speed of light they travel on the small like the surface of optimal infinity and that's what we're interested so if you the ball the definitions of of of of the so the medium definition that adjusts all the previous slide have something like an equivalent will have corresponding definition at vanity and recalled this the ball on definitions that goes back to Troutman Bondi Bonnybrook Metzner and Saxena 19 fifties and sixties To define mass energy and momentum was at that moment well more ,comma constraints probably you have to put there to make things work on saying work and famous theorem of course and you that mass is positive and so this was shown by Shane and yelling 1981 and then also by Edward separately and then in the bond a case of this was study by Sheehan Galilee so 1st of all energy what's not another real a good way to talk about energy so energy controls the curvature 1st of all that what kind of energy so we used about Robinson Tanzer quite often is 1 solution to 2 this problem so we can think of the Bill Robinson tangerines so this is a defined as follows so this is basically the 4 Cabrera intensive field but it's basically if you look at that at the it in a while tents so you look at the Wild Tanja of your space time and basically the Bill Robinson is a quadratic in your wild isn't it has a lot of nice properties it is actually a positive in the sense that if you plug in future directed Time-Life vectors so you get this possibility property and it also satisfies the questions to his justification for the use of hospitals and anonymity .period out so this was used heavily in their work by Cristobal inclined 1 on the arms Makovsky stability so this is really the main object 2 to control the curvature and nowadays in many other people as well so let me also hinted introduced a currently Kate disk you know the better Robinson just introduced so contract is now with 3 future directed calls electric fields and the opting what called the car and in this case so I'm going to call his check but my car and then I can apply some divergent theory on abounded domain in the space time and I can also then find what is my career to flocks so if the only got here is just to the boundary of this city's contained a portion of an all-white surface so we remember that come with some of find 200 molded to Fieldale then the corresponding boundary term is indeed the curvature flocks through this at the surface and they can write down as like this interval so this is the courage of flocks we define like associate to some small electric field so these are some important on objects to think about it and often when you would like to proof at the controls of the Bill Robinson Tanzer actually controls the curvature and from there we can go on so maybe back to the crucial problem 1 tiny moment so if you look at the initial laid off of this type so I have not yet told you how much the case you need sector for certain questions so if you look at Cristobal Klein amongst some incontestable idiot-proof and Sixers generalization to Einstein case so the initial data hasn't this kind of form so you look at initial later where for a large part Chibok has a like 1 over month to and what are here the 1st heard and a case like that to optimize the house and their corresponding decays also for came with some some leader regularity and the marginalization of these results will have less decay so I'll have to give ottomans 1 year and some less regularity and I can ask what's the unique in order to see some interesting things in terms of radiation memory effect so it turns out I mean here we get all the information and in fact Christodoulou used except in the last chapter of he spoke with that surgery to derived his momentum memory effect of gravitational waves so not only to find that the exists as unique as a solution to have a perfect description of how the asymptotic behavior looks like and so my case while I do see some things it can compute the few things but in order to get the full picture this kind of hidden in the of the carriages very low very slowly he said the services
will also be used to have use of the power arrest I mean that with each them even if you to go down by 1 powerful are you running so here are some of the right down to the evolution equations to constraints and collapsed just to give you an idea so I mean can then write down said the answer questions the coupling to constraints and evolution equation suffice to my Smilax function so I have equation to evolve the metric chaebol are given by that this right inside have an equation to involve the secondment from PIJ and so here this is actually or from each space flight slots on the right side and if I choose a natural affiliation meaning of the trace of K 0 so the constraint equations reduced to this set of equations and an additional half an equation for the lapse of a maximum pollution here to look like so it again so you can put this into hyperbolic formats said try to work with them now let me also
mentioned just briefly the stability resolved by Cristobal climbed 1 so this was a big breakthrough of course the question is can you find any Minkowski space just a little bit together a global solution which is counted as a clear complete and for all time so and be well this is 1 very simple version of the theory to state its fully need a few more pages but you could say that every isn't publicly flap initially at which is globally close to McCloskey and globally closes sense with waited Sobolev space is at a crown on certain of norms have to be small enough so this gives you a solution which is a complete space-time just tending to McCloskey infinity along any students so I interesting enough so again so that I was just a few weeks ago I spoke to few physicist like Come strumming oratory and beaten now out talk all about the critical Kleinman space time so they aren't there at the moment very much looking at their looking for solutions which are exact is no approximation anything and it turns out is said before so the last chapter of this book gives the most precise information about some interesting physical space time of course you can always try to I mean do approximations etc. but it's interesting also to know our if you may relax go I do something else a little bit how this does your solution really look like so I I mentioned internalization by well in my generalizations there is you don't really see as much of radiation you can't really read these things of that well there's many many people that I'm not citing everybody who have worked on related problems so let me just say this is actually interesting for us because we get
a precise description of Noland and that's what I would like to stress also OK maybe well just here on the blackboard I already explained this but we usually work with 2 types of foliations in in this setting so 1 is given by a space like cut the surface knows called Sigma here which called H on the blackboard and also on all hyper surfaces so we will be interested in outgoing not surfaces and fight for simplicity they look just like microbes on my fly well my think again to remember I use this on the maximal time functions trace of this case police Suro each space slide is a complete money and had surface with corresponding decay and then I'm interested in the intersection of these so when intersects the space like an all-night the surface I get of course 2nd well as an object just a few more fit was fear in this case so as to you look just at the intersection of the it will be imported what Watkins also let me just introduce a little bit more so I mean I can look at the knowledge of field generating no electric field ale in the outgoing direction on a cold day laboratory 120 injuring direction and maybe this will be the 4 later and this will be the treaty and then I can complement this with the awful frame just on the surface space here and I will actually be composed now components ,comma curvature and my genetic components with respect to such affiliation where this is given Eleanor Bosnia OK and well let's look at some at the bonding has said something I mentioned before this is going to be interesting for radiation safety look at 1 such moment the surface quality you in our space time and let the time go to infinity so we would like to see what happens to local quantities which is from locally what's the limit up so I can define what is called the hawking mask so "quotation mark local it's like integrate over such as sphere basically St you want the trace of car increase of cried bottle what status so maybe should introduce it as well so "quotation mark 2 important objects when I draw this picture again that's when you see draw often so if I have talent and a lot of generating vector fields in all directions so I can say well up here have the kind of X and Y civics admiring the tangent space here is of his ass so this is going to be given by Cheney nobler exhaled as wise as the 2nd fundamental form in the mail outgoing all erection that look at the FA Act version of course doing the same thing with the team going nowhere so this order corresponding 2nd fundamental forms in the mall direction and interesting for us will be the shares this will be to trace those parts of these objects sold by Act just use this to call this Christmas party and that of course have to trace of these objects which comes up in the definition here so 1 way to write down the hockey mask as an integral over "quotation mark locally with such and such a surface is given like trees kind times trace Khyber lots of the trace of these objects and won't given like that now the question for interesting space times that not be too specific at the moment so this talking has actually tends to ask when they go out to Noland vanity so if you look at the blackboard over there or here we go out to Nolanville unit so that this is ongoing going to them what we call the bonding so it has a finite limit for the space times we're interested in not what its future no infinity the heuristic we already introduced that so I will call his on lost so this is defined to be the end points off all the future directed melted 6 along which August Infinity I cast just the tip of the polity of operas has to do with the function you taking values in so can think of function you being parentage rising your no infinity so I our sits up here and is depending on you of course so each knowledge surface has a finite bonding that's of course if you space-time is not that they nice enough this could blow up or not be defined but in that case as we started here dimensions of finite and the OK now the body-mass mass measures so what does it do so it measures in this sense the amount of mass which is remaining in an isolated system as measured at at Noland at the given recorded time out here so basically well out here is maybe CU which is hitting but this is no Infinity for once optional hyper surface so I have a certain amount and we will also see the bonding mouthfuls formula by saying taking the relays with respect to its function you so this will give us how much radiation actually has happened so here it is so if you just look at times and matching we don't care about other fields just pure gravity acting so many of the so-called mouthfuls formula and again this is something this terminology is also used just copying the terminology of the spoke with so this is also in the sitting there With the arrived and well understood so you have a bonding mass loss for Milosevic to the derivative and on the right hand side to have this CSI object is actually know the limit of 1 of these years so this Chirac's With that the bottom version so you say don't want to see you elect to go to infinity undertake delivered so this is what is called exercise a few years and correspondingly this other guy has a limit as well so let me just write it down because it will show up end so this is called a signal that so this art objects no infinity so I explained this already easily have the 2nd from other forms of assistance traces .period which gives the Shearson holds that portion begins identifies I told you that already so let's skip offered now
what gravitational radiation like gave set up of wealth the space times when looking at but gravitational radiation is now a fluctuation of the curvature of space Thompson is based on a lot of structure in terms of Ramon encouragement and proper and so when a gravitational waves travels from the source so it's changing the face the curvature of the space time that's like the packet the makeover a one-second traveling through and that's how we think of and I would like to see what we can learn about but he also let me also mentions so-called memory effect for the moment so OK we have heard about lying and Lygo looks like an L-shaped it's an L-shaped detectors so this is the same distance like here and for 90 degree angle and if he for simplicity is you know that the gravitational waves sources coming from the 3rd perpendicular direction so that it acts like a planner ways out here so the wave will sell Of course it hits from different angles but in order to simplify the discussion here so let's assume a comes from this direction perpendicular so what will happen is that this Masters will move in the play in the play because it's like a plane wave when it hits our detectors over so this Masters will move and what they were able to measure in this experiment is of Ilidza interferometry you measure the distance off let's 8 masked 1 from syrup must 2 from 0 this is where I actually knew you have of being splinter hero by laser interferometry measure the distance and of course the Kyrgyz changing the distance the space-time is changing and it reflected the displacement of this test so here the test massacred think of their floating on their to so you see what the genetics of doing by looking at the test so OK so they were able to measure this instantaneous displacements during the time of the past by of this way and I can ask what happens afterward this is not to be detected but let's hope in the near future may be so what you think is in many in most cases you would think a well things go back to their children 6 us before everything looks the same now there is a prediction called the memory effect saying not all this will not happen it will happen it will be that the space time will be permanently changed and spare the test masters will be from permanently displaced for that matter so this is what is called the memory effect of gravitational waves and while there are various studies many people have in the meantime worked on that slippery maybe just a little bit about it so in a linear rise version of our nearest theory of Giants equations the 1st people to find such an effect works although which Empoli in the 70's and then in a fully economic problem but setting that was Dmitri who use the last chapter of the cost delivered book of his and searches to Armenia plug-in the known as politics also are true for a March leader 1 can show so he formed the arrived from all that TV rights the so-called nonlinear manner so people call the linear and non-linear and it was up to the 1st effect was supposed to be so small never to be detected but then this effect is actually large enough it's also small but large enough to be detected hopefully and so people always thought this is really a linear and non-linear thank you Of the same on the same effect as memory but on the turns out some of Garfinkel look that bad and we all know we found that these are 2 different effects which have to do with the leader 1 the call regular so this has to do with 1 portion of that while curvature so the elect 1 portion of the electric part of the wild curvature changing over time and that the novel or formerly nonlinear effect has to do with feels that really go out to know Infiniti so things that change finished so that 2 different things and we found is also in the linear Maxwell equations to which women's well not displacement but kicks so anyway so you can ask Well what do well it's magnetic fields to on neutrinos so they will actually only adds to the 2nd effect which call no effect when many people have worked on that there is work by Bible ocean Warburton skittish cook for and many people I'm actually probably missing but it's interesting that recently from ignorance collaborators ,comma Farm also took up the idea of this work on memory and so they have the idea that the memory effect is actually part of the triangle where they look at Ward identities and be super translations and a lot of people looking for memory effect other field theories like the idiocy of tea and anything you can think of so it seems that there's something interesting but maybe not but that's just the beginning of understanding of what this means of theories OK well there are 2 types of memory and let me know Hong Kong back to the relented curvature if you think about but the decomposed is a little bit and kind of lay open the structure of regulation will would like to show so against 3 denotes just from the 3 is just in going all of nectar feel that right down here and the forest the outgoing electric field so if I'm just in the irons and vacuum equations for the rerun curvature is my while curvature and I decompose here the Kruger components with respect to this foliations now the most interesting part to remembers his Alpha Bah this part which does like 1 of are the town minus you can think of like you it's 1 pursue Square Square so there's a particles like 1 all are and this will be the interesting part of the forest for radiation so and then you can look at different space times and see what happens with this so if you stay within the so-called critical climb on space space-times see well the album or parts has a limit that small infinity call capital any of you and this is a symmetric trees free took a very intensive field on the sphere of Norman finished all the components also have certain behavior I'm not going to look at that but if you change the type of space time and you do another caution problem to really understand fully what's happening at no infinity so that you can do more to settings and so we found Garfinkel the also came up with a different method so that the most rigorous way to study that is of course to do the problem for each set of initially Idaho well this can be cumbersome if you're just interest letting radiation so we have another method which is an approximation perturbing the wild curvature which is gage invariant to actually for more general space-times also look at graduation Noland Of the many years just 1 short notes on 1 of these methods so crystal Kleinman actually introduced an interesting set of an interesting theory looking at but the elliptic equations on such a surface and then propagating along all over space like direction so if you look at the trace of course I was in a fight with the parameter s in Imola erection so this is given by the right and inside of some of the sheer squared with a trace of client Ben business now just anything if you plug in the 9th and not annulled Florida this comes from the intermittent tens of the right to decide and this constantly have Einstein Axel commissions this will be a component of the electromagnetic field so In general when you look at the gals equation on things look like so I can write down the curvature of this intersection Michael St you in terms of office follows a half to trace ,comma cardboard right inside a half-year program of the shares plus WI Colchester a component of the wild curvature and contribution from cheese will have either a quadratic since you're trace of car I want pure component of while other than the health of aboard bars the 1 with the stick or contribution from the TV contract so I wondered if I look at the milk adults equations in these notations so we can define what it called a mass suspect function or its conjugate so see as just a portion so this is some the again of this structure and that we can write this will help of the gas equation then we can actually write the song masses that function like this and now if you look at the mall not see coke responding conjugate cadets equations so they have a form like that so you have quite a quadratic of Richie coefficients and right inside or a derivative of 1 of those years another contracting all are pure component of With curvature but not born not the worst and contributions from Chile if any so am I already told you about the limits may landscape that so the limits against al-Qaida hacked and cry had this year's have limits like the left here at the moment now the structure let me introduce a little bit more about the structures of the side is a component of secondment formal portion following is a component of the while curvature and teachers energy momentum so that he can write down and end is just that would normally into to enter into the inner space like slide so you have the structure of the equations when you look at the propagation of a kind heart so you have a quadratic again in sun or a derivative of such a PCI term and me have just curvature Quartey itself and ETA had actually kind bottle with their kind are so similar for the other equations if you look at this
time of equations antique limits I would like to stand happens all Infinity's silicon's multiplied by people are all that is needed and then we see on the right inside so palpable or has a limit which is called a so we get for these equations begets limits actually for many different space times this is actually true that the behavior of known fitted easily have this behavior between the shares of the dividend that everybody's respect to you and once you're here is a given related to the curvature no Infinity which is the 1 who all or part of the the no energy radiated so we can say Well in appearance much case so the energy radiated this integrated from minus plus Infinity would make the show this year so if you are impure Einstein vacuum you only have this sheer part but if you add electromagnetic fields have a contribution from the electromagnetic fields or what portion and if you have that's a neutrino static and model by some novel slow you will also get a positive contribution to this energy which is radiated away on business so we integrate this from minus 2 plus infinity at no infinity OK here is maybe
a theorem just generally I mean in the eyes match case you can just forget about the aspect so what happens if you have different fields and Johnson much in the iPhone equations so we can say that this city most of the shares of this car I have over here so this is the limit of this shares so it has limits and lost and about minus infinity for use of this has limits and this will be related to this permanent change of your space time after gravitational wave has passed and well as note b any tenderer function which depends on the fields with their rights decay and he will note any lower components of the energy and stressed cancer well that's what the theorem who says Well we also have some function fine which is a solution of the following so we have F minus its mean value as bought over a sphere at infinity and this stranger patient means just atmosphere at infinity and then this different Sigma Sigma minus is given by this equation so in other words all the regulation here or what is right the way this energy comes into the difference of this shares at Nolan vanity and this is directly related to some
displacement of has so many let me skipped the ideas of the proven to something else and you can prove this you need some up to investigate some hot systems locally and then go to infinity and take the limits of these and see what happens there L well now it can be shown that for now we're still flat space times so the permanent displacement if I write this down again you remember from beef or so here the test masters of this type and the gravitational wave comes and travel through so now claim it and this is not an this is just a dealt the claim is that the permanent displacement is given by this right inside and you see this right inside is this difference of the shares at MOL Infiniti that I just developed in the theorem and interfering this is linked to the energy radiated away which is then sourced in the pure gravitational case it's off by source by the limit of disks Eisele of this novel of 2nd from form here and if you have extra feels like a 11 1 in the quarter neutrinos they will actually active OK and so there's also something called the ordinary memory that's what people thought was too lenient and before so and this is now a completely different things so that no memories what the big portion of this permanent place displaced actually and while many
here very briefly how the seduction of relate experiment have the well by really studying the call problem and looking at the NOL infinity so we can really derive a 1 of geometric or analytic information but how does this not reflect anything in Ickesburg well if I set up an experiment so let's think of 306 in space time and you them by Dennis 0 1 until such suggested genetics on which my spirit my particles of floating and Welty is my future units time active field sitting on a chemist hero then I'm looking what's happening and what's happening so I look at the statistics I introduce an often Wilfredo fields like constructive frame field along the Gamez 0 genetic and well I can then set things up so that I can measure nicely what's happening with the distance between 1 and 2 0 0 2 1 0 now I can go out on the specific circumstances I can replace a geodesic by Jacoby equation and to derivatives left site which give me the curvature component on the right here in tracking contracting with X L an interesting enough so depending Everything is with the city this curvature components so you really need to understand the crucial problem based on looks like to really understand what's happening here and the interesting thing is maybe let me also add now an opportunistic of foreign all fluid so basically your team you knew some positive function here KIT J. nor Kazan electorate and looking at the twice contracted beyond candidacy justify plucking and also that now flowing so we have shown that well this now for you also contribute the ice equations Ranulph which reduce just to this equation here so the space-time which is given by some constant times the image momentum tend to wear not Felicity now if you look at the portion of seasonal flu which has led to the right decay behavior is so it decays like 1 squared something you and would like to understand how is not that we want her to related to this portion of the mole fluid which comes in from the right equipment site on the question well know that the real and curvature can be decomposed into the trace less part which is the while curvature and we have richer curvature and skated for retreat and so well if you plug everything in and look at that lets worst components we find serious just at the elements so we find that well if you look at that while we have some component in the pure while curvature but also through the Richie coming from the mold fluid if you look at the major component go a look at the corresponding Einstein all fluid on the right side so which component replied in the worst components and the USA no affiliation again and when you plug everything in and go back to the north element Bob notation can see well there's 2 things that happened 1st of all I have is alpha blocker Richard which is still at 1 of alright and define the limit and write it down like this and have this component of the Nile flowing which goes like 1 of our squared the Well wait a minute so we have 1 of law in the courage and was lost for adheres it's not contributes this is lowered so it's true for the instantaneous instantaneously displacement so when gravitational waves traveling through so this is indeed a low order for that but it turns out that for the cumulative effect after which the memory this is actually the same order how does that work so but let me write down looking at maybe 2 more minutes so the right down the 2nd derivative here Jacoby equation again from the right inside and out this is that the notation for their courage complimented no affinity that went over or square are compiled so now we have this nice relation between this year and the curvature and we know also that the leaders of basic ideas to 0 when you this very large so this means that if you have passed so here we have just plucked and this means that if you integrate your wanted to substitute this means that the velocity will be Sierra after the gravitational wave pass something happens but for large you the velocity will be serious the crew back to rest by substitute twice here this year's subplot that into this equation and let me take the fall limit I get exactly what I told you before namely this displacement here which is just permanent displacement is given by the difference of this shares which again hidden behind the right insiders this energy which is ready the ways of what exactly contributions from space time what extracted contributes to it too this summer please display OK maybe I should say 1 more word just at the at the end so with Garfinkel study also cosmological space-times the question is if you are in a cosmological setting up these things still bear 180 interference Hollywood Francis the cosmological constant coming so this display any role so will look at at Robertson Walker plus sectors so what positive cosmological constant be found for the visitor spaced out so can write and metric like that so the desert space-time models still basically inflation period of the universe that's 1 way to think about it so in that case we found in the stated that well there is a factor wobbles was are steered Hubble radius which are the Hubble constant so which is given you on and enhancement of his memory effect to the memory effects multiplied by this factor so becomes bigger something it's something similar maybe a chance through that to the very end but something similar you can say for that if the law wkt so this is work-in-progress for writing it not but for SLR W which is modeling basically our present universe you can also see that this memory will be enhanced by such a factor so many I stop you thank you but the without this is the last thing we need to have a friend who try out yet so 1st of all let me maybe make this a little bit more precise so if earlier exactly so if I have just a circle and that the waves hitting from here so what it does basically it's kind of stretching and squeezing tried so it's like a planner so this is 1 of the arts and it can think of is is like well I'd take points on them and now that the claim is that with the memory so you would have this has a certain well displacement in this direction business by 90 degrees in the other direction so this will be permanently displaced and then you can actually compute what the hell would be they were in the area would be a clearly feels more is not really a thing with computer From wireless this is also the only 1 who can refer to 1 aspect of the problem so memories at low frequencies and also what they saw in let's say if you look at it that the strength of the signal based solid develop the land that's like 10 to the minus 21 of what they saw no there's a paper brought proposing to to measure a memory would like go that just came out the few weeks ago so they propose something like at the audit into the minus 22 and also all the noise that you can think of is at low frequency for light goes so it's probably not the best way to look for it but nevertheless so some people are proposing to do that and the they have and so some kind of a filter they hope they put there so I don't know they are improving instruments so that they could yeah right but I mean that the weather absolutely I mean the where where the place to look for that would be the space project that I did not send easier has been working on and I think the Pathfinder was launched by the European group and so spaces would be this easy well easy technical details and now model with all of those out so this would be really easier to see their also with the frequency and Illinois he said the level of the Member of this memory yet so actually we thought about the bill looked at the pure Maxwell equations which Alenia of course and there there's some if you take let's say Jackson and take 1 of the foremost play around so you get easily you get easily what we call what is called the linear logical the ordinary memory so and this would not be a permanent displacement but you could think instead here this masses are not chartists test masses following the geodesic but what he would do for the Maxwell cases you have charged bounces and you would get an overall but velocity after the passage of an electromagnetic waves so a cake and so we found that ordinary thing and we also found all cake which is then again Baker and corresponds to the non-memory here actually and that's the Purim classical Maxwell equations Corsican now ask What about QED well 1 thing seems to be there the other we don't understand yet some we're going to hear
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Titel The Einstein Equations and Gravitational Radiation
Serientitel Trimestre Ondes Non Linéaires - May conference
Teil 02
Anzahl der Teile 21
Autor Bieri, Lydia
Lizenz CC-Namensnennung 3.0 Unported:
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DOI 10.5446/20768
Herausgeber Institut des Hautes Études Scientifiques (IHÉS)
Erscheinungsjahr 2016
Sprache Englisch

Inhaltliche Metadaten

Fachgebiet Mathematik
Abstract In Mathematical General Relativity (GR) the Einstein equations describe the laws of the universe. This system of hyperbolic nonlinear pde has served as a playground for all kinds of new problems and methods in pde analysis and geometry. A major goal in the study of these equations is to investigate the analytic properties and geometries of the solution spacetimes. In particular, fluctuations of the curvature of the spacetime, known as gravitational waves, have been a highly active research topic. A few weeks ago, it was confirmed that advanced LIGO detected gravitational waves. Understanding gravitational radiation is tightly interwoven with the study of the Cauchy problem in GR. I will talk about geometric-analytic results on gravitational radiation and the memory effect of gravitational waves. We will connect the mathematical findings to experiments. I will also address recent work with David Garfinkle on gravitational radiation in asymptotically at as well as cosmological spacetimes.

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