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1/4 Gravitational Waves and Binary Systems

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1/4 Gravitational Waves and Binary Systems
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This crash course will review the theory of the generation of gravitational waves, as well as the theory of the motion and radiation of the premier expected source for gravitational wave interferometric detectors: binary systems.
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constantly Actions dynamic inner integrators time loss argument part energy permutations derivators mechanisms mathematics estimates sign Geschwindigkeit square negative field eccentricities systems relation generation moment infinity physical staff expansions Hot meetings indicators binary trees gravitational waves orders sum Right flux spacetime meter Drall functionality non-existence momentum factor Sequel Newtonian high resolution similar Mass student distances events theoretical fractals powerful second period Wellen Propagation Bewegungsgleichungen causal Lecture/Conference Formula terms Average form symbols differential equations initial conditions addition multiple law commutes basis total applications approximate formalism calculations Factory Universal optics Gravity waveform objects band family unite
testing principals functionality Actions Einstein field equations Transformers diagonals time loss dimension powerful subset derivators period sign Wellen mathematics sorting Lecture/Conference Formula terms operations square form equations eccentricities systems area Blocks coordination indicators Hot approximate numbers means curves flat spacetime gravitational waves logic Planwelle combined Gravity Right metrics spacetime
testing complex Actions integrators time directions contraction plane conjugation Geschwindigkeit different matrix systems exponentiation job coordination indicators gravitational waves real vector angles source spacetime functionality free group Transformers component real transfer distances Coloured rules events Wellen Propagation Lecture/Conference terms Formula symbols equations conditions addition incident Planwelle combined electromagnetic waves Gravity objects coefficients Superpositions transverse unite
complex Actions presentation breadth directions time water part orders of magnitude dimension report mathematics plane matrix square Elastic circle field extent systems exponentiation flow expansions indicators domain Tangent means gravitational waves angles real vector Symmetry orders phase lips sum Results spacetime source point non-existence functionality sweeping component student quantum field theories rules polarization events theoretical powerful period Wellen goods Propagation Lecture/Conference terms complex operations level equations set conditions multiple scale surface basis lines Axialer Vektor approximate Spring flat spacetime combined calculations electromagnetic waves Gravity mathematical object objects band coefficients epsilon transverse localization unite
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presentation Actions breadth directions time ones contraction loss argument dimension orders of magnitude report representation derivators plane notation sorting Geschwindigkeit analog field position moment Computing infinity physical expansions nominations connections real vector Symmetry normal Right sum cycle mathematicians Results source spacetime point geometry polynomial functionality factor Einheitskugel link varieties component high resolution EDMS student events rules product period goods Wellen specific Densities Propagation Lecture/Conference terms Formula operations differentiable operation form symbols equations conditions multiple scale eigenvalue bases model structure independent basis lines rate applications Sphere subgroups Kernels potential formalism topology combined calculations Gravity objects coefficients table family unite
building Actions transportation integrators time contraction loss argument part energy sites representation derivators mathematics notation matrix square field representation theory systems area job sigma Blocks moment infinity expansions maximal indicators Hot sequence Symmetry quantum sum Results flux source spacetime Bas point meter current functionality Drall free group momentum Transformers high resolution tournament component time machine similar EDMS Mass transfer Electromagnetic product elements training powerful Wellen goods Densities Formula terms Average decomposition operations form theorems equations standards multiple volume basis total Deltafunktion numbers approximate signatures invariants Factory Gravity objects game table coefficients epsilon
behind the bar and you and you and if we I think this is the Bloomberg and the new before starting and I wanted to did you that's out there 2 books supplies and every command or on the topics of today there is the need to book which was finished last year by year unshirkable at the age of 91 and cheese the only person I know we discuss with lunchtime because she was the first one to pore over the existence of solution lunchtime secretions contain obligations by waves and this is an excellent book and for students but very responsive but with physics and that mall on the precise topic this is the 1st volume of the excellent book of McKinley majority McAnally is now finishing his 2nd volume on the more experimental parts of the the way this is the 1st volume on the series OK so so here I have the reminded you that in 1916 acted in November 1915 Stein for this spot where some help with from here and found general election a formatter coupled to gravity and just for people with minor accident never seen that sort of thing remind you I will use a convention with the signature mostly plus the letter urging him denotes the minus the determinant of the metric of absurd or Romanian space-time so the definition of the curvature of space-time is given that the metric tons Europe this combination of partial derivative of the retreat which is that the Wichita connection that for me you defined the only man to answer and the construction which is the which Einstein's equations on the left hand side of the Einstein answer which he minus one-half of the trace of our times the symmetric and on the right hand side of Utah as the functional derivative of the matter action with respect to Junior with the metric which is the stress energy-intensive matter OK just some conventions because I so the aim of the calls is to go former from these equations 2 To the recent results which here which is that of it's set in September 2015 the 2 Lygo detector arose sold but noisy signal and this signal is in excellent agreement with theoretical predictions and the 1st theoretical prediction was an analytical prediction which was done here at cheers in 2000 with the methods call effective 1 body that I will explain in detail and our dismantled can't predict I can't predict not only the did do this is the way form muted so I will explain what is a gravitational waves from scratch with his reforms this is the way form as a function of time in the affected 1 body the method there is the 1st of which is inspiring when the 2 bodies are still parents separated and then they get closer and closer and then the method defines the merger and then attached to the signal what happens after that to block costs coalesced together with cheese and I will explain how you you consider that the so-called rolling down so this was the 1st prediction 5 years before numerical wanted to exist than 15 years before experiment existed to detect them and but this has been literally flying after all the after the existence of numerical rigidities in addition and now means the recurrent 290 bowls our will appear with understandable when I go and the Global Fund a sum of 1 1 0 in the States defined in full whole version of the 51 with reform which took so extracted some information from numerical relativity and and here there is a sketch of defied that after this kind of tuning of the United takes into numerical there is an exquisite agreement between numerical into an effective 1 body and ending in these things have been used to extract the signal from the noise although the order of the stages of extraction and this is the things that have been used to extract the masses for instance the fact that you said the masses up 29 56 is because you have fitted to something that continues to 2 parameters and then you maximize the overlap so the aim of the course be to go also known it may explain some basic things to fall entering some technical details so after creating generosity in November at 19 15 of in the enjoyment of 1960 in a few months later lunchtime and worked out of the structure of gravitational waves I mean everybody knew that generosity was a feeling that was generalizing kind of Mike's for integration so there would be ways in them there was no question they would be with the question was what was the structure of these waves and what was the amount of the generation of these waves by matter actually the 1st paper about and 1916 is for it as a mistake which is not a minor mistake and the food to the correct answer essentially was given in in 1918 and in this paper also Einstein gave a former Los giving the amount of energy emitted by moving masses which depends on himself and find their way into the portable movement of the system and that but what I want to to describe it is that starting from this work it is after the 1980 and that's uh all the modern methods that led to this things will develop and so I would mention so there are 2 types of works that started in the 1970's 1 work was about the dynamics of the dynamics of binary system so the costs will be about what is the motion of 2 bodies witnesses and that albeit around each other by other gravitational force and what is the gravitational waves the meat and what is the back reaction of the gravitational
ruminations on the motion and then what is the final of the signal of mission taking into account reaction and the coalescence of this tools look to objects when they are blocked and I will not discuss the case where neutron stars so that the the history of the of the dynamics of 2 weather systems in general activity started in 19 17 when Lawrence and lost computed the 1st approximation beyond Newton's law so as you all saw their original 2 bodies should attract each other by 1 of square you talk but then you have corrections due to general ITT corrections proportional not only to the square of the velocity divided by the visit applied but also due to nonlinear effects and this was 1st worked out by Lawrence and lost and then later by lunchtime in from of mindful 1938 the next approximation so this is the approximation University Square the next approximation of the overseas fall and together with overseas 5 was the 1st worked out it's 2 and 2 . 5 PM at the 1991 19 meeting to buy mentally there therein was here and there and by me and then there the next approximation the EPA which means you over the the 6 wall the 1st computed by Younis sky Schaefer and myself in 2000 and at the time when I said we had to finish this work so I thought this would be enough and that was complicated enough in the meantime mathematical existed and computer-aided calculations were made possible after this I really felt that was the end but to kill Janofsky is very enterprising and and then in 2000 fault so Yassky Schiffer and myself we got to the defaulting and funds will sign the fallopian approximation which is respect to the part where and 8 and bend these the ingredients are important ingredients for the story there is another so this is the ingredients about the Smith said the equations of motion of a binary system another very important ingredient is the generation for formalism and the and the best generation generation fund and deformities and for generating gravitational waves and the best formalism is the so-called Pinochet them wider formalism which started only little bit before I came here when I worked with her blow share was my PhD student and then which was important when bad idea I came here in 1989 when just arrived so I will explain what is this formalism and this followers and is what allows 1 to compute thinks they're sold all of these things is that post Newtonian work so this users post Newtonian theory which means in a general physics language deprivation theory so is you use expansion in the small and the small parameter essentially he's the velocity of the orbital velocity divided by the velocity of light but in the coalescence of 2 black the ratio of the overseas becomes 1 of at the end of Mildred sold the small parameter appeals OK and the airport emission gets bad before you which marriages and and this is why I will explain that there was another family here which is to go beyond perturbation theory to use his summation technique and and this is within this thing that 1 of the most efficient and accurate thing created in 99 2000 was the immunity DUP formalism that I will describe in detail now so this is essentially the plan of the cause now let me remind basic facts at the at the lowest all there that is to say things that we're known in the 60's uh this fall this worker began and the basic findings I'm in mind is the work of Peter Rose and tells and my fear was that to Matthews 6 4 and to of the terms alone in 1964 now what they did is they use the lowest although approximation that Einstein I've worked out in 1918 that they will describe which says that if the wave form the gravitational wave emitted by a binary by in the system but that's about binary system is proportional to the 2nd time that narrative of the quadrupled moment of the system with the system moves along with fault of the moment varies in time and then you have a gravitational within metres and then Aristide gave a formula saying that the energy the flux of energy into that Infinity is so this is a material may be effective tool that Einstein got from conceptually good to try to include many an oversight the amount of energy per unit time limited as a flux at infinity in the form of competition waves Is is there is a 2nd factor with Newton's constant staff did not say Geez Newton's constancy the velocity of light that forbidding the interior of the quadruple monitor the system square immediately or over I will use Einstein summation convention each time there is an index finger so the square means you put this thing twice you some of the increases I injured between 1 and 3 details and my fear was added to this the loss of angular momentum in the system in the form of gravitational waves at infinity and and they got a similar formula which contains 2 geographer 5 did want to symmetric they've achieved symbol which means that sit on educating press 1 if I is even permutation of 1 2 3 and minus 1 if it is odd commutation of 1 2 3 and then see if you have a the 2nd time derivative of the possible moment and the filter and everything with some investors contracted this way then they uh then the joystick really assume that if you lose energy and angular momentum at infinity this energy negative momentum will be taken from the system and their fault that the binary system going on entity called story of 2 bodies and going around the center of mass going like that 1 going like that and the other was going like that that's the energy and angular momentum of this system
mechanical energy we'll be joining them out of the system because of the as at infinity so your rights that you have losses of energy equal to the fluxes at Infinity I should say that this is because of this indirect reasoning that the use of the system loses energy to our additional with therefore the system should pay for it that people said that the observation of binary Purcell's that of check are indirect costs of the existence of additional addition actually work with Nathalie we have done direct calculations because there will not ascribed to that of the interaction in spacetime of 2 bodies including nonlinear effects of gravity sold something which never spoke about waves into that Infiniti but which states that it takes a certain time for the interaction to Polegate from 1 body to the other body because public it's with the velocity of lighting and incentives so it's it's a with it's only covered with NBC television in the calculation gives an extra term integration so motion and then from this excited actually you cover of the soul and you don't need to your think this is good this is directly linked to the fact that got into public it's a diversity of light so then pigtails and my view is used the relation between the AIG of the system like for instance and the angular momentum of the system that just like that the energy of the system is really include when I have to Brussels to do so my major axis Of the VAT be between the 2 by formula of this type of a similar formula for general momentum then you write that the the time derivative of this therefore the time they were of the sublimely Draxes should be acquitted 2 miners this you compute the average of these you compute what is a probable movement of a binary system at the end you get explicit formula saying so to form alliances the 1st Formula you should know that this is the the 1st formula is that the orbital period of the system decreases secured only because the Beit new ideas by kept kept players fear law says that the total the system is affiliated to the orbital frequencies only gonna where only got you have to buy divided by the orbital period so it's the orbital frequency basis of my major accidents so this is the famous 1 to where the masses power 1 of the optical frequencies power to end of the summer Midrock seizes power and then this relates to the period to but here you have this summer majorities which is related to the energy loss of energy so finally you have a change of period and again Natalie we have shown directly without using this reasoning by reject dynamical calculations that indeed the propagation of the beauty of the velocity of light implies a change of the period of a binary system which is lower stop estimation is given by an explicit for 1 9 2 5 5 by the way the reason why there isn't 1 of the 5 in order formalize is because the speed of the Gallatin is to its initiative to end to express 1 where with as sequel to his 2 times to press 1 so it's 5 that's why there is always this 1 overlooked whispers 1 which are like spin fractals of them along the old formula and things so then you have the factory Gilbert frequency here to fight over please but peacefully put peak times to Newton's constant to the five-furlong then there is the policy of the masses divided by the sum To the power over 1 3rd of and then you have a certain function of eccentricity here which is why because of binary system in general as the eccentricity of this good the 796 before deciding about 100 years prior to the final of the list no this change of the orbital period of a binary system due to the with obligation of amity between the 2 2 objective has been verified experimentally to very high accuracy 1 part in the thousand in similar binary Purcell systems so we know this is real and what are the consequence of the 1st consequences is that there is a minus sign OK so this is saying that the system is going faster and faster than can period it is diminishing which means the 2 objectives closer and closer because they're close are they have talked more each other and then they go faster and faster and this is why these things I've been going on for all 5 under Muni as you know if you take it to what these separated by a very large distances they go wrong but the orbital period decreases the get closer and closer so they go faster and faster and at the end when there are very very close the Qantas refused to make a final objects from the topic this is what has been seen this event happened would billion light years away from testing to be able to to estimate the cost of the timescale here is 0 . 2 seconds so the these are the last which had been seen in the last field beats it goes so fast after the velocity of like the must and have to objects of kilometres to go alone so it's bond once again for 1 albeit at the end and that the following existence falls and mutineers but we couldn't see from the other no what I wanted to say is to win so the 1st thing we know that the orbital period with decrees but Fonda's or the formula used to determine the change of angular momentum will imply a change of eccentricity or so and then when you work out the change of eccentricity appeared as was able to solve implausible Mission allergy eccentricity depends on the period because you have you differential equations like by the team is a function of eccentricity and period and you have disappeared over the team which is a function of the function of eccentricity here but if you take the ratio of the tool you have a differential equation for all the years of function of peace and actually was a to solve separable and surprisingly that the solution is both simple and surprisingly complicated because the exact solution at the lowest although is that the square provided by 1 minus the square to the 19th 6 times and 1 closed down 21 over 3 or 4 the square to the power of this is this power is always really surprising it's the power 145 divided by 121 and all and simple things generate Apollo like that and this thing is moderate constantly bands which depends on the initial conditions is proportional to the period to the power of 19 9 actually these
exact powers on a very important what is important is as I say that the orbital period decreases it goes faster and faster and therefore the eccentricity because there is a square area decreases even if you start with logic simplicity it would decrease and at the end the acceptance is also what you get is that the square is proportional to a power of pity and therefore the period decreases increases the eccentricity indicators which means that the U.S. secularization actually this improvization whether effective if you start with systems which are really take the losses of enrollment 1st make them and enlarged and white systems they stay wide but they're out nearly circular OK and then after being so they get sick 1st and then the simple thing gets smaller and smaller of a spiral of OK and this is an important simplification because 1 could do on those theories because at the end of the albeit this circular so it simplifies many things and 1 can compute in all their things because it is a six-year-old care I go in the space in the rear of the stink so I've already shown now we go faster some technical things so let me solve now gravitational waves and a cleaner rise automation Soul II Soul of Einstein's equations acting arise approximation this means what 1 assumes continued to cope with the mystery is the flat Italian you denotes the Minkowski metric viable minus press press press the diagonal got most of what I will do will be in full dimensions he Avatar and Einstein's equation in spacetime dimensions you say that in any dimension these is one-off here on the left but when you put it on the right there is the dimension which comes in it's only in spacetime fall that's on the right side of also minus 1 of K but there will be injured call for most of the time and then you have a perturbation In space-time of this girl metric and you assume that each new new small compared to what they say is this is press 1 minus 1 so it's news dimension now you take this formula you compute things you probe this here is very easy you have DH Uganda tendency at doctoral to lost all then again not using your age you could hear man and which he contends to derivative of each new when you right explicitly the lean arrives the understand answered I really do like you go and you find there is a 1 off because there is this 1 of there is a 1 here and then you find that this is minus the number version of Virginia None of Russian means flat that abortion OK so it is great and you knew you knew where the new means the partial believe the repeated with respect to its new new then you have a 2nd term with the same sign minus choose New Age where age means the taste at the new new official new OK or at other public ritual for the test summed and then you have to terms 2 weeks later Page and from new but knew of each of them knew where the index on finds ways to buy the flats and Minkowski metric Nowlin arising out of the soul it is anticipated uh you look for solutions of at approximation off the pitch equally or because when I have no matter this form of angina Gration Stanley it's which flat and now old there are 2 ways of of 2 ways of solving and that's what Einstein 19 16 in 1980 we want to find solutions of these equations Popeye getting as waves in empty space a simple thing is too limited to look for plane waves that is to say USA LK let me like Mike 20 questions look for solutions where this is a function of X I haven't amplitude which is the principal and then I have a K X where K X the means to on the explained OK so it's mindset if I decomposing space and time means K X minus 0 magnetic effect so it is a phrase sparked off a wave with a k a special frequency and all my garden circular on time frequency and you plug this year so immediately you're saying that this is equal to 0 tells you that you get the question so this gives case square yet Wecker square km in a community who can you can you you know where we think we 1st thing so the block operator of disk a square in Union plus the KNU continue where a denotes the Tracy so I'm going to use it and you knew you you knew uh and then you have the other terms which are minors can you can In new London minors the new kid on the block new loan Everything aboard to 0 OK but no but how do you solve that 1st let's assume if square which means this OK uh where different from 0 I could soar for new knew from this aggression so I will get a new new equal 1 of square of what I would get something like ingenue I have continued be Europe can you believe you we have a new user these things these press 1 of of this OK I will get a combination of this type but now we have to remember that Einstein's equations are invariant under all called in a transformations and therefore fault realize Einstein equations are also in variants on the infinitesimal coordinate transformation has taken a long flat space so if I take an infinitesimal coordinate transformation changeover of coordinates from its new to expire New this is a precise can you find that the effect on eyes gravity means that the makes changes by each new minus the new exciting new miners continue set in the 2 1 and it's I knew if I am in the real also plane waves that is to say if I do according a transformation which is also aware like this means like for sensitive sign you is accorded to know How did I do not it OK OK if it isn't it time you exponential hiking then this thing with the miners I came you so it can you -minus I continue the tenuous and their therefore the change
of cold that has the effect of a gage to what is called this transformation on each menu which is of the type can you sort Victor operas skin you have the same effect of but we have fallen here that if case where is different from the old ingenuity would be of this type therefore anywhere which does not propagate with the velocity of lights the use of this type which is a coordinates with the it's not that we're aware of and therefore necessarily the winners must propagate to rid the velocity of light so this is a simple way of saying we must have formed an additional risk is critical OK but now once you have kids square "quotation mark
0 otherwise this is not to where it's an artifact of changing coordinate system the same equation is mall that is to say not only public it's with the velocity of life but I most of these conditions to write this conditions let me define let me define would define fingers given any tenser I define it about new like it's continuing minus 1 of the trace of dispenser we've at that time said commune with the minus 1 OK and in particular about new new things in a new minus 1 of traces of ingenuity times at if I use just the definition these 2 terms simplified Anelka's Griquas 0 and then I get the condition that uh 0 0 is a color to the longer times it bought new longer which she's so I get the 1st condition for gravitational waves his career with the old and the 2nd condition for the 1st conditions as appropriate with the veracity of like the 2nd conditions than many its at times our sanity conditions because it says that the contraction of the amplitude of the modified with the vector of obligations space-time is sold plans are subject to the conditions but now I can do even OK so this is 1 way of getting them plane waves there is another way that I will not do indeed temperatures just mentioned which before solving Einstein's equations directly he I have not chosen in a coordinated set by the way they did something surprising here which is that even if you don't fix the coordinate system at all if you are aware they must satisfy this condition which is characteristic of a special where would fixing coordinates which is called the on gage sold the other polls are is to go to on Monday coordinates which is too to change the coordinates systems this time to solve an equation Fox I knew of the type something such that the following condition is satisfied that the longer Iginla new loan according to you you know you can show in general that it is always possible independently of saying I discuss so waves OK I can fix gage that they use in Nanchang aggression by this condition when you do that you recognize that these 2 terms actually because here you can like this is twice 1 and then you put 1 out here and 1 of the and you find that these 2 terms actually they well I can absorb distance and therefore is this is the old these 2 terms disappear and then in the summer Nicole it seems that the ruling there was the chief concern minus 1 of Donald there the question anew this is analogous to considering Maxwell equations if you take Maxwell equations uh you know the new if you knew you could you know in vacuum with new do you mean you minus the new anyway so that we have an exciting new Maxwell equations they become minus the number of immune plus the new of the divergent so it's very similar to me if I go to the lowlands gage and by the way a party this Florence is not the famous Lawrence it's supposed to be launched without the Newark Lorenz gage where this is 0 then he Lorenz gage Maxwell's equations become but this year the Commission is minus the velvet according to the sources said there is a source and nothing if you yourself and you have the condition the new injury would you go you see what I'm saying patients of a similar with 1 index more tests of this incident when now what is the structural gravitational waves here I can still do something because I said I do according to information I have this effect I said here I
can I have already use quality at information to simplify things but I can still do another coordinate transformation because if I do segment called in and Continental confirmation and said thank I knew in addition to the 1 I did before and did prime satisfied is a wave-like satisfied "quotation mark zeal then I will not change the condition before you can check that this thing is invariant under according a transformation if the quality of transformation public it's with the velocity of life so I can further reduce the dissolution and a nice way to go to so-called moved to T transverse trace date now let me write the gravitational waves in transfer stresses gage this is a gage where you impose that did the 0 0 and the zeal I components whereas always a time of eyes the space called of the old and you only have the spatial components of a plane wave amplitude and when you do that and you could this formula this thing simplifies you find immediately that the trace aim was being 0 and therefore the special because is so once you have imposed this special interests must be 0 and then the special intense His China's versed in the usual sense that is to say this is a and it is orthogonal when contracted for spoke to the direction of publications space of the gravitational waves so this is a plane wave In transverse process engaged with the condition caseworkers if I know maker superposition of these waves for life we are in France we have before you the Rams so any thing can be expanded in four-year intervals and so I think combined general solution of Einstein's equations no it's new and then in this village a few you find that you intend to gage this yours your bike by convention in 0 0 as as well as do the will I compliment only components that exist of the IJ component of Chase is 0 . 4 and the the questions of propagation of 1st of Kansas senator conditions this condition becomes when I consider for general functional makes this 170 and this condition becomes that only of changing times to conclude 0 so this is what the wave of additional ways looks like noted in Korea can use this but so in in practical terms what it means is that if I consider wave which propagates in the direction came then which would be the Z axis OK when we take so much since then I have a plane so let's say this is Zen-like place then I ever played orthogonal to this which is ex-wife playing the gravitational waves of only components unit of the ex-wife played so in that sense the Web itself is only his transverse and sold it in that case was case by the unit Victoria Z then HIG something which is now it's it's a freebie free matrix in charging us too especially in the sewers and it is a function of the entire Z in that case for a plane wave if I look at the company then it will have because of the time the sanitary conditions on those things that containers and index would be 0 or I will have only X X Y and X Y but this is symmetric and the traced the chase was busy which means that some of these these and 0 so it means I am 1 thinking I have minus same thinking and then I have something in there so I have and then I have the exponential of white distrust explicitly accused of using the claims process complex conjugate because we're interested in real solution lunched integration ended the four-year modes of complex which means that this wastes as components of the hearing the the function of the event when Anglicans days but it thank you said is that the miners reached 1st critics this symbol Austrians in 0 0 2 7 0 means person costs these are so close to the capitalization of about additional ways this is analogous to the fact that when I have Maxwell and Atlantic where I can go not only in Lorenz gage but individual is always yield 1 can further improve this heavily AI which must be times were sold sold and then when the wistful by getting the that direction and electromagnetic wave as only the components the X they wanted and 0 so that like the 1st thing and so here I have the X the 1 0 0 and an exponential decay expressed sympathy and physically this means the acts as a function of the instead the employer functions inside and you'll sold got additional waves is very similar to an Atlantic waves in the sense that there are only 2 independent function given the Whipple by getting some direction have to arbitrate the time that gave this 1 and this 1 like entities transverse like there ,comma integrates but there is a big difference and the big difference to seats you need to define the inner-city amplitude this is a name but we are going to understand 1 and it is inner-city it means taking a linear combination of the Of the 2 wanted with a complex coefficients OK so innovator allowing it isn't you defined 8 in the city amplitude 4 plus 1 0 minus 1 in the city and of the definition is any acts and because of some convention when there is a unit to put a minus just a convention uh so you take the a complex combination of the 2 constitutes if you do the same thing for gravity you define and amplitude pulled referred to instead of 1 which is this is this amplitudes OK with the same combination so it is a of the gravity-wave minus process I and the other thing and now the the fact is that so here we have 2 mathematical objects which are waves living in the plane X Y all told to the publication of the way I can do still I can act on this wave by alteration either either in the place I can help leaving seeks the directions and that I cannot take the coordinate system altered the wave shop passive or active when you do that if you walk takes the quality system by an angle you find that this thing the article might take 1 wheel because I so there would be "quotation mark inside OK but now I have taken complex combination and by the 1 of formula that I will have an exponential so you find that this is exponential plus or minus I think that the
initiative the foyer calling it isn't with equivocation 1 here why if you do that for gravity you find that in a personal minus 2 4 gravity of politician is experimental personal minus to 5 feet off the coast of minus 2 so this is saying that the the mathematical objects which people against the the gravitational wave in the transverse plane when you all day by Michael thank God it's denounced by an angle to two-thirds why and they with terms by an angle one-fifth public and this is in general or if something by exponential act as thick as that you say that it elicited as OK so this is just a simple rule without doing Qantas edition of the thing saying that in any city of gravitational waves is espresso minus 2 widely in the city of Atlantic waves in espresso minus 1 the subject In an apology in the orders the time of work carried out when Parker was giving lectures was an assistant with everything going with the flow of water from the book so this was the case city of the fermentation away now I want to briefly the mentioned the fact that 1 of 2 things Glenn Avenue elastic aquarium finally the object that represents the wave is made off affect all In the Plains transverse to the propagation so the ex-wife Plains so in this plane I have called in its system explains and then the object and talk about User vector and now if you compute at 1 point of space this vector is sum of exponential I'm a and things like that so it's a sweep of Tate's OK it is a sum of co-signers and then you find that it takes a while which of these 2 components that all in general and the lips if you think that we are part of the 2 it would have to elicit is to complex numbers so you have a wave is described by 2 complex number was 1 in the minus-4 in general readers too complex numbers of units of the same magnitude on not proportional to each other or whatever you have it's so this is saying this is an elliptically polarized every 20 graves if you do the same thing for gravity actually you have essentially the same thing because here you can consider the object so so here this summer we can say this is a complex plane so this hour all in the complex tale is a expressed by the White River gravitational waves you can do the same thing you can take each trust but expressed meant what it magician XX it's the XX component this is minus its white wine which is mine as a child and is it each ex-wife so you think could take these things this is a a quantity this is a real quantities so now I have a complex quantity and in general rule it is a combination of the coastal and inland to and then again you have an elliptic you have it's Sweeney In this spring soju a simple thing which is but trust special cases like this if for instance 1 of the 2 the city opportunities you'll you find that this week circle OK on the other end that if the 2 things are proportional complex number to each other you will have a line you without the elites with generating line so this is already nearly polarize waves this would be an assisted only for alive wave and the general condition is annually to keep alive so it's exactly the same thing as for pulling it isn't I'm envisioning at least except there's this to footing won't let me also mentioned something let me go back to space here I was using I was in space To say of the directions k propagation excluded said and then have the plane X 1 issue mean Koskinen you can say OK a wave you is gravitational wave you described by 1 of and then intimate will so it's specified by the 1st the character can you which is the way vector of the wave which witches on the Net :colon so I have please NOTE OK and now if I elastic wave I have amounted to attuned a new which she's all told the Lord to begin you so this means that I must consider the plane the plane October low to the right quality is also tension to the right "quotation mark if I take the view that the IPO surface all too little to another Victor Oconee wasn't as quickly you is isotropic as 1 the French in the old days uh and uh the plane of the to Kenny was also the plane tangent at plane tangent I am in the press 1 the mentioned you need to of one-dimensional and now I know that and electromagnetic wave is described by some polarization vector which is October OK with epsilon he knew he could and then the electrolytic wave would be the new recalled that said some of the students in the Newark recruits and kids in particular I can't take a complex I'm OK and this is it do they have to Bechtel's Alta land in the fall space-time dimensions so this is a three-dimensional address 1st and a set of extols here there have been the victim of a kidney result of to itself and then I have 2 more effect also I can't choose for instance Ibsen anew these I can take a basis and also to was he won in the 2 which on October capability 1 was you'll like it looks too so I take to the tolls or too little to Kate and such that they have unique magnitude for once could to square and that they Cobol known to each other it's just on business OK now let me take a particular polarization vector which correspond to a circular polarization so if I think this combination it is a security report lines and accommodating with now why am I saying that I'm saying that because if you do the same thing with gravity with the same polarization complex polarization it to the new you Yukon construct also of who could only polarize gravitational waves which is simply an altitude coefficient times it shouldn't you the kids In space OK and then therefore you see that's the polarization dancer of a gravitational wave In this simple thing is just the Paul ducts of 2 whether to integrate and this thing players all in the same period in the fact that in
insisting theory and no also in quantum field theory commodities like the square off at a community than actually you can see it already at the level of often waits now to finish the 1st however they would be a 15 minute break let me say that he I have done really naive things really arise 1st older it's more of a wave propagating around a flat because the mathematics is easier we are interested in these wave here which has been upset last September I was admitted to the billion light years away and then tumbled in the cosmological space-time is that to pass all galaxies and things like that in the current space-time this is not flat space between us and this thing so uh the full fury of gravitational waves uh he asked to combine 2 things locally in the system or on the sources you can use flat space-time calculation but then when the wave propagates fire in spacetime you have to use under the technique and on the way to do that is to be sold to compute ,comma additional waves in enclosed spaces is to use the W me all high-frequency approximation which means that the general gravitational waves you can say it a space-time which he is the son of a bygone space-time which only contains makes role of frequencies like something which varies on cosmological times all galactic times and and then you add to it a small wave prolific worship Epsilon which will also of accepted by method that is it also varies with the slow and scale but today's referred fast time scale so you introduce like the fascinating phase which means you take a large 1 and and then you look for a solution of Einstein's equation of this type as the extent of the call as an approximation in interest powers of W K approximation and as and when you do that actually it's it's settled because it depends how small it is excellent compared to the Omega if only got his 1st solo there in 1 over effects on his 1st order mother 1 of the only guy changes the back the action of these wave on this but for this I refer you to the recent book of Yvonne shook because indeed this is something that even shook it without herself worked as a research paper longer ago and this is carefully explained in her book so all of that to stop no-fault break OK so now that you've on shook Award is with us on the personal 1st pulled mathematically the existence of a solution lies exact last nite creations including the men of dependence which means gravitational wave propagation and will discuss with lunchtime in 1952 in Princeton now let me go to support so now I wouldn't a more technical details in order to be able to compute things to high curiously we need actually to go beyond reminded you that Einstein worked out the lowest older so-called quadruple approximation and even the public or approximation was done to lowest stalled in posted expansion but anyhow and agreeing to report you love them before thing left us something things to do and that and now I want to discuss the question of much support extension In eyes gravity but the and surprisingly surprisingly that does In the time demands that the 1st for the correct discussion of multiple or expansions in Linas gravity was done by banner idea and myself here In 91 His ready 91 following work that I will explain at length later by blow shade and myself I'm not another going to karaoke 89 the so let me that means would because I've always consider if you open a textbook and even for calling it isn't if you open the textbook which means that Jackson and use you look at Monte Pora expansion of rental antique fumes I consider it a mess I cannot understand what is going on there are strangers objects you don't know what is going on I during the away and and it is in the four-year demanded is that you don't even computer what is emitted by the time time-varying sources you have to four-year transform with thing and then you have this and functions is actually the work that was done by us Starting in In 1990 Wallace in the time domain and that and use uh technical to order that Yukon at suggested would be good which to use the the the symmetry tastefully uh Cartesian tenser quick presentation of multiple expansion so let me explain this what is expansion multiple expansion is an expansion in he do similar presentations of so don't let your mind broke if you know what it is and I will explain what it means soul and I we're really issue for instance and if you have a field which is just a scared of field but we really do the effect of fewer than the tensile field which satisfies the the gracious of this type which some sources local binders full power salsa but I consider the field outside the source so that it satisfied that on their fight could 0 and I am interested in the recovered waves generated by the source but the important thing is that this operator here is invariant under no alterations in space creating Vermont on wall but as the source usually defines itself in June but you can define a Norwegian somewhere in space which continues to exist time and you are interested in quotations acting in space now what does it mean that this is invariant in space it means that if I have a solution 5 of this equation 0 in the white inside and if I define it's a function of time and space and that if I defined the new functions 5 all 5 refrain index on a couple of TV economics which 5 off and I minus 1 events where are is it's an all too little matrix it's alteration in the usual sense that's active alteration directive which what takes points in space you make it acts with universal that it is a good implementation whispers of function is just a technical the point is that if this function fight is a solution of the fight could this new functional 5
4 in the automatically is again a solution of the same immigration OK uh you remember that he I'm really alternating the solution in the sense that if I look at a function 5 points accepted even the end of the cold modulus effects as a function of angles and so I defined text called on the way the moderates and is the unit on the unit's sphere so this becomes a function of the unity of my for instance the use of low here and it is yours with politicians means I really wanted to wear the Brumbies can't so it's a different functions now if I consider if I continue considering this the fight as a function of the politicians and the media blitz on the unit's failure this becomes a function on the units we have now and function of the units can be expanded the name Phoenix the series labeled by an integer and going for zeal to infinity to press infinity where such each building block here is such that when I act on this by this operation I looked it then these brought here by on a navigate in a finite dimensional space we've had mentioned to impress ones the fact that there is a vector space we find dimensions such that all the solutions were inadequate politician down the narrow combination of some business so this is the a and and now let's be practical what he is a member of the NPC is the irreducible implementation of the hope of alterations and viewed on the street what is a practical way of the parliament's rising this thing functions so if I take over the functions of a in the space of the function on the unit's and is satisfied square I am in space when I put their knowledge inside of the dimensional retreat in space can now there are several ways of quick presenting its but a nice way is the STF when it means I can't ,comma tries and by tenser with disuse what I want to to a contractor degrees and I want and I too and I can't solve it is something which is the polynomial off and for there in the unit vector constrained by the by this thing and the hat means that the Spencer the symmetry so we find a way to for changing this as it does not change and Tyson predefined take any ties on a new pair of this as it is 0 so it's cold stiff symmetry transferring it is already be and and now if I acted with the defiant act with alteration on with put it as an action on the method acting on offense and what does it mean when I actually what it means is that I'd change I and I won come again converted the action on this on this but there ominous 1 so then it means Spencer becomes the new dancer Hi and where as prime with aliases is obtained by the alteration matrix .period on June 1 that said ,comma end militants G 1 Jr OK so attacks on the transfer of the usual alterations in the space of these already dispenser of use of mentioned to will press 1 now which 1 can see you in various ways and this is I as I will explain later this is equivalent to the wire lends the usual remedy things but the Wayland confuse things because for the reasons I will expect now a simple way of understanding why symmetry jurisprudence potentials of I'm carrying the game was a nice way of Parliament rising this is to consider for instance not that a questionable the Laplace equation of classical fight with 0 and let me consider the general solution of Laplace equation in board for instance for instance the unit born with laggard ideology and but for some conditions on the on the boundary of the ball and we don't ask with integration while that led the former said this is edited equation so the solution is analytic actually and because it is an entity I can write that fight .period X separated from the origin can be expanded in the power series there'll series so excited residual respect side that they've achieved 4 or 5 plus 1 hour 12 scientists Jake the idea 5 plus 1 of the 5 and free the exit of the derivatives and things like that now let me immediately introduced a convenient uh notation that we introduce referring to Rongcheng which is that's when I have a tenser with a leading this is small and you don't want to like always I wanted to let you look at uh the only information which is useful is that they are allegiances In their fault the convention is that you like just a capital and OK if she died that he sporting is you would put the capital P so is just a symbol for saying that there adding users and then depart of a vector of the same vector a at times you just you a N N a K is just a contract notation but it means adding this is with them so it's a general Einstein summation convention now use some of emerging diseases In an and now this thing can they return as some of and from 0 to infinity of 1 of reflect on their land because this is the roles for acts as the end of 5 0 which means again so experiments X 1 X 2 up to and as Indian means of imputed derivative but but you see that this thing is the sum of a N of XM some objects with framing this FM which is this divided by race but this subject is symmetric because it is a commodity derivative and it is priceless because if I think that softening business as this is the accords the old it is your so this is this way I have proven that the general solution inside the Borel fight can be in the the general solution laggard ideology can be written this way now I can also rightly some of I can like the victim next is on the victor and book therefore this thing is not to the public and I take the modulus informed the vector and as and 1 and 2 at a time when I won and I too and then find a different kind and now I can override the creation of that class of fight with decomposing that class as original delivered plus 1 over all square the angle that ability of acting on the angles state fight on the street that's the usual formula quantum mechanics of the plants where the separator candidate and either as 1 of the square the of squared the ball as 1 over on the square are all as the duty twice because 2 over geolocated Saudi to to this and now I know that this is a general solution of plastic would you also when
you when you heard this information here you find easily that this pose that the angered operator acting on an island yes you have to respect these this subject when I put the Haddad means unitary interest-free if I contacted symmetric trace freed by something else in projects the other thing also symmetry to respect OK so this thing is also known as White symmetry to a 3 to make it usable then I find that the underdog fighting please just by saying this is a solution abrupt last I find this when the pilots and Olympus 1 and then now that Apple than this I can ask a different question let me find what is the general solution of flat plastic with between 2 shells not swagger on the origin but between our long enough to I expanded its so what do I do if I expanded my right side equaled some of the off a N N III decompose it was administering this better when I me right I can write it this way as a function on the unit's sphere I can decompose unanimity for a series in this symmetry traced people section of and this is equivalent to a wire led the composition in the striker three-kilometer OK when I do that and I use this equation I write a lot classified or I get a differential equation for affair and you find easy that this differential equation as to solutions 1 of which is the winnowing out to the press and which is a good on the origin and the other 1 which is 1 of the art to the and press 1 indeed L and minus any post 1 of of as as press 1 "quotation mark Island Press 1 OK you find 2nd the equation for the ball rule in case you know that so the conclusion is that the general solution of that classic was your between 2 shells will be on this type where the subject fan of art which signatory face reading so it's offensive adding this is and it's a function of but we just so I just argued quickly that this function will be the sum of her after the island 1 over up to the Empress so this thing will be a certain symmetry tree Sweden which is a constant finds out to the end process and not there the thing which is the to the art to the miners and press 1 because now before I had only the 1st term because the other terms Singirok the origin so this is the general solution of that classic with your between 2 shells and this is equivalent to collect me like this now let me considered the general solution of Laplace equation uh in vacuum which is gaining at infinity now I impose the other condition I want something which is a regular at infinity now I need to cancel the stomach because this some cause at infinity so I keep only the other atoms so just to just 2 I understand what well talking about so this evening .period and a general solution of plastic accords 0 which he cares that Infiniti is this infinite series over other in the series on the sphere they they converge the weak conditions of the French ability like being sued to on the CIA number 1 that discussed with my Simon was intent that when we're developing this site asking Can I write this series with the convention Barry and they said the adultery you seem to be something that anyway since is don't get too much but it's a weak conditions what does it mean respect to the octopus 1 brief I wanted it explicitly it needs I an object for equality which has no witnesses tho it is a constant misstated the 0 pulled off bus 1 plus Ivan object with 1 index and I also square cuts of an objected to induces which is a signatory countries face NI NJ let me liked what he is what is an IG acquired that evening and nite NJ the signatory to so it means an eye injury I must take its symmetrical wedding so what they need is to take away the trace so it's minus 1 the tragic wasn't square good so you see that the 2nd term is 1 of the but plus etc. but this is the usual metaphor expansion know this is the moment for 1 of over this is the guy the idea of vector or this is a part of born and objects in victories we refer to something there so you weak over the usual multiple expansion but what can I do even better in the sense that there is a useful formula which is that d f repeated that evidence of 1 of our OK uh without doing any calculation what do we know about this 1 over all satisfies that class of 1 of our articles -minus 4 by then of distance outside there we dealing the zeal of ideology so it is money if I think that a video of a nominee function it is again harmonic can solve this thing will be heard by dimension lines is 1 of the art to the press 1 and is monarch therefore it must be 1 of the Dominique functions have beaten them and indeed should do a calculation you just need to find what is the questions and from it's minus to them by yeah I never understood why the full 15 years when people write the was minus 1 to the power of looking at the 1 it's the sign my nose to the public and that the means just to simplify efficient so it's minus to them To and minus 1 of them 2nd what is it about factorial a double five-volume means you you do on behalf of the things like the bullfighter 5 is 5 times 3 times 1 OK reflect all your 77 pence 5 incidents that you don't make make half of that it's it should be a house factorial but it is probable for that uh and which means the generalization of this with me in users you think the product you take away all traces divided by onto the expressway OK and and their fault model a change of notation by absorbing this factor in the existing candidate and some of our car slightly modified the which is used to mitigate risk we did come 1 after but now we have something which is very easy to understand because finally the multiple the general solution of plastic was outside the source is simply a them repeated that haven't even won over our which is evidently on Monday Kennedy's signature interest we we sometimes are informed and this is symmetric persuaded the answer is projected to be symmetric transfer so this can be generalized OK parent too yes so now let us apply so these outfield multiples in the sense I said maybe I am discussing in fact I am now in discussing this equation not that class equation but quest on which which means the source and I put a minus fall by because of the number of that task so I prefer to the minus for 5 there and then I don't have to put it together and what the Green function uh let me remind you that the Green functions for the plastic question is that is 1 of the explained his wife in the sense that this is a to minus fault lines that have functioned three-dimensional explains why can't uh because of the 2nd soft was equation as porcelain discovery along the here France by writing that 5 x is equal to the integral over space so now all that need hold something I have also and I consider it .period inside the cell cycle while I consider it .period X outside the salsa I compute dissolution fireworks outside the source as a function of the on the other density in the salt so I just started the pursuant to which she's roll of the wide divided about 1 of the X minus 1 OK but no this is a function of X and Y where and why the only GEC I think diligently if text is outside the source said this distance why he is smaller than 1 compared to explore ratio of why the distance compared to its and therefore I can expand this in a convergence series in Powell's of whites in Natick function and encourages industry domain but when I
do that's what do I get actually so let's why did I have tables the which says that 1 of our X minus 1 that the advantage of this notation is that thinks becomes very simple 1 over X ante that look simple 1 over X minus 1 can be written as some over half of my because there is a minus to the end White to the at the end of 1 of the this is as if was you know 1 valuable and you were expanding 1 of X minus 1 and I accept that this means pencil failing this assistance of repeated that he would admit this is going insist the interference with and valuables and if I plugged so you fight inserts Monaghan's this thing here I will get that fire outside the salsa so now I take it outside is to some over and from 0 to infinity of minus 1 to the end of a factorial crimes and fat there is only 1 of the Act but the city's modulus of X is what I call before this year this is the end of last so it's the same it's exactly what I was writing here this thing here is this thing and therefore the coefficient I have informed is what I was quoting before the field multiple moments what twice as the field in which a voice but now it is expressed in terms of the source liabilities or I wanted the result is that the Futenma to border which is symmetric and amount these when I when I consider I put the fishermen then you find that this is the usual well-known for the feet wide inside also because the Copeland this article but supporting the finds white you see this is the simple formula that we all know the multiple moments is deemed to have over the source of the density times the pizza Y 1 for the dipole why I fall the probable or why I Y J but they take away the trace is just that ization for years and if any "quotation mark 0 I have what little mask if any 1 added that poor woman out of the source if any were 2 of the probable moments of the souls of Now let me make the connection in these expansions and the head of the Russian putting on and wire lends generalizes things actually there is something nice that I discovered when the reading Maxwell uh because in the Book of Maxwell is as very nice insights about physics in general and multiple expansions and then why are they called multiples because actually what makes Windows it says let's concede that the general solutions of the plastic wood 0 of which decays at infinity and the East saying the general solution would be obtained as follows I take on the sphere of each vector OK uh you take a 1st vector that you call the fall Paul again because of what is the bible the dipole isn't infinitesimal displacement between the miners drowned in the press structure it is a direction OK let me normalizes direction something so it defines also on the unit's sphere defines where dismantled pierces this fiercely defines a point on the street but now let me take other points so I take points on this field which means cycles the 1 . 2 points as they can be you like those but we can't put the normal later OK I take any of them then Maxwell explicitly says that model OK I take this factor I take the 1st electoral I made the point that the attends a product of an local icon tried this with the head of the University of 1 of our OK this is money will consult this is this is also mnemonic function which depends on what's OK 1st if I write it unity is 1 of our and 1 just by dimensional analysis therefore on top of this I will have a polynomial which depends on an adventure but a polynomial in and the unit vector on this view because the fight take on those Bechtel's coincidence at 1 point so I think pitched the same Victor this polynomial easily John polynomial where the Z axis is the regional axis OK but if I take them I'll be trying uh I get a polynomial for instance if I take adequate to find that appeared to fall to victory if you do the computation is 1 of also we did 1 end of the scale products due to end -minus you wanted to to scare a product of the 1 and 2 which is the contraction of the potential product of the 2 correct also plants one-half of the and NJ minors the idea and now you see immediately the the minus 1 this is the road from putting on is that they are in the ZX it's OK but you never generalizes on polynomial which depends on Bechtel's on this here and now if you take but now betrayed let's take all the called the vector was wounded simply so that they are on this field for all sphere iTunes and OK 1 2 and how many did not do I have 1 point is to call the notes that the firing the sphere if I take em points I have to to ask continuous parameters here I put the magnitude and faults will carry the normalizing with normalizing factor OK so it gives to a press 1 of laboratory parameters so he'll have an object which contains to whip was 1 of the Tory bottles which is our among which satisfies the full thing is on money and then actually uh this is the the most general solution of us agree and that's the way it might slow the Wallace family ties include multiple and that's why they're called 2 boards because they are made of boards and boards working on the now what is the link between this air and the usual so I just said quickly that if I take all the victuals yes so let me the what is confusing about violence is that when you have multiple expansions like that you parametrized then by symmetry trees served without of writing the stance of this certainly is a general trend certainly aspects of Kent a as basis to impress one-dimensional because there are many conditions but you don't need to write it on the basis is like when you do geometry you can say I have effectively you don't need to say I have a VAX BYU presented with explicit number of OK you work abstract and the STF Portuguese like that why if you say no no I want to express things in the X Y Z axis OK then you choose a business this is the 1 away business is to choose a business all the possible role of symmetry try sweetened cereals are at rank and and the way you choose the business's the following is the Mets when when you take any borders on the students but what are the sports there operating in the following way 1st you have to choose exists yet in wide use later in space so they said OK let me take XYZ actually now let us call you said that you could use your just for fun because it 0 means it does you always city in the sense that if I may quotations in the ex-wife claimed as Alexis doesn't move so as the presentation of the subgroup of operation the two-dimensional plane this is the eigenvalues 0 OK wide if I want something which is a good idea vector although alterations in the XY playing I need to have a complex combination which traditionally defined like that so it is no all yikes personal mind tied 1 we relative to this adjusted normalization and this is also normalization for historical reasons no the plus and a minus in in the same order as as you were before the real in symbol of so this is this is a victory as a complex vector which is known by the way his titles are such that there eyes a public square but still the formerly you can concede there by using this formula that's why land is obtained by taking the report you used this formalizes the you'd you define a solution of replies to questions by acting on the it's because mathematically they say is mathematicians you know
would said vector is a differential operator but and therefore I think that Dole's acting on 1 of successes uh so you take the victim EZ you take it sort of me check this story is that which I called 0 you take kids and minus and times and then you take a the director of the press 1 that you take em times which means that idea I write this formula and that he had in mind assembled his vitals are on and M of them are you press 1 taking this formula is defines a solution nominate solution this thing these whole along to so let what this this subject when I act this way so this differential but even mathematically Edwards said I act with the press 1 and as a vector or as a differential operators and times and the 0 lower payments and times I acted 1 of and what the rules require efficient that I don't want to right here this thing is why lemme effect of fire look at the usual spherical monarchs and that but in explicit form wonderland means it is an object which is a polynomial reef adding In and then OK with my notation so the conclusion is that the usual wire they defined and how many of these do any of that I have to help this 1 because and goes between miners and and personal OK that's the usual things I by the Whittington the formula for positive and 1 it's a formula with minus 1 forming it's even the point I want to make here is that the usual violence if you open the textbook I don't know if it's called the Book of Kilimanjaro and you will find definition away lens at these things can be returned as polynomials like these like John is will remain in name and contracted with the victim was informed their fault defiance was seen its rates this thing is a signatory traced and that the sect of Tuen press 1 therefore Devan and and enervating between minus 7 so I have to Empress 1 pencils symmetry tryst with themselves and the Spencers is the basis it's in October mold bases with respect to the integral role this week off the set off of independents a a symmetry jurisprudence are in this to impress one-dimensional space and therefore the conclusion is that what is confusing with when you expand multiple expansion in 2011 is rapidly 2 things you are expanding intrinsically in terms of public presentations but also computing the components of this representation specific basis which actually confuses the the thing more the than I find personal now now let's go this was not desecration of and closely question we are interested in Einstein's equations but I have explained this at length just to say all you use this technology to compute the multiple expansion 14 yards variety secretion and it really paid it's a surprising fact that this was 1st done by me in a bad idea here in 90 but did not exist and correctly in the literature and not in there was something which was well uh and and not to the offensive and this why do explained this because the smutty port expansion in gravity these are the basis of what is useful not only in our gratitude became and all the calculations in the promotion of Ohio for many of them that there have been done using the nominal versions of what I'm going to prevent additional viewers they sell so what one-time explains for a direct last equation the press 20 questions not only new fall and finds creation OK but 1st we have to do it for all these political decline golden although for the logistics and get a few Batista said let us consider the creation which is the warranty idealistic questioning creation which is this one-liners for line as students became so you take the National in spacetime you take the source which is especially with specially compact support so it's a bit that persists in time is that it moves in space and it certainly does not move on average space into 0 here 0 along that you can't now I'll do you and we are interested in the recounted solution you can delete bounded solutions he is obtained by 5 is that we haven't been function convoluted with the salsa as where the only time then function as a function of 2 points in space and time so that we probably function is a function of a point in space .period yet which as a client component in the space component and I'm going to to have you so this is a field .period This is a source the function the white wine is equal to as everybody knows is that found potential it is the time to extend the white minus 4 with other thing X minus 1 divided by the divided by its minus 1 so you have the usual queso 1 over all but 2 of that we call the propagation of the velocity rights we are in the false the mentioned this is when now but now and the fall what you want you have a source here so you compute this convolution so you OK the table will be over there might coincide with an avenue toward him but now underway expanding multiples what does it mean when people but the number of multiple it sweetened the father of pressing question it was a simple multiple where sum all of the repeated special derivative of the busy ,comma Nick function 1 of our work can I write immediately what will be the the form of a gentleman to port expansions yes because outside the source file as a function key events could be assigned Over and from 0 to infinity I can't conventionally it's convenient to keep the same disinterested numerical factor this way takes away 1 of Ltd Wal-Mart I want know something which she's the solution of bank would you and which is now emitted by the source so it is like a function what is the basic functions of the TNI are emitted by the source we all know this is his effort minus obviously divided by all of this is that we call the wave emitted by line became so if I consider this and I take on suspicion that he would you this this is a solution of them Americans the old if I put adding this is here at the resolution and therefore I expect that the general solutions will have multiple sources defined by symmetric trace thanks also F is something we've had in this is and symmetry week but it's also a function of time and I liked it as a function of T minus overall therefore it is a solution of what and no 1st you pull over that this is told that is to say that the retarded solution that you could Butterbean function can be expanded this way but the important thing is that you want to compute what he's family in terms of the source of that which is the analog of the formula here which was saying that the X-Men was the usual multiple became integrated of assault but now everything is fine independent so it becomes a nontrivial calculation and this is the calculation which was filled on by revulsion and myself in 89 and 1 gets a simple final answer but it's not as simple as the 1 above days in small of a trip to attend and the answer is fair it is a signatory to answer with this services which is a function of 1 valuable book and time and actually I would like 1st of all which is the same as this 1 FM is an interval of salsa the the finally now just for simplicity I called y I call experts X is 1 OK am in the sauce all of a sudden function is about which is a function of TMX Times exit so is the same formalize this 1 I think of multiple moments like the quotable moments in
interest but I have a certain source as saying OK which is not as itself so it's not what the source into slightly modified but what is as biographer of this formula and as about which is a function of TV and that's these equal or to an integral to between minus 1 interests 1 off over a no exterior viable descendant of that at the end of Z. The Times the sources of the 1 that appears here that we solved case
that's computer to time per se because Z 6 of X where the IEA you have an average between so you out instead of computing things at time t the device aboard deity primer deviates from the by something which is between minus all the Sea Empress there the physical reason is that indeed if I conclude this at time t and 20 cm high and some federations around which is the time travel time in the soft so I need to average overtime or time in the source and that the current Delta and then he is a function was in Table of that is equal to 1 but this is an average the internal that dialogs that desired between minus 1 plus 1 is equal to 1 why is called that time if you take as a lounge this function that these looks like a delta function in the sense that for large and it is concentrated around 0 it's still very mingling between minus 1 plus 1 the area here is 1 Virginia is concentrated around the city with and the explicit formula is very simple plays a numerical coefficient which is to persuade the bullfight going on devising to to the end 1 5 oil and it's just a quick efficient plants one-liners that square to the power times the step function of one-liners Square which means it is 0 outside this interval so you see it decays exponentially when Z is different from 0 by minus a script to the public and when end is and so this is I this is a very convenient formula which allows to compute explicitly that the quality the statement to pour moments for the for the massless can a few and it is this formula that you you apply for solving and you want to solve essentially the same equation except that you are going to find investors for instance if you want to solve Maxwell equations you like this but you put an index instead source is Jr the current fall victim if I saw Einstein's equation I will have to win OK the new new OK but then for each each solution I can use this formula but In visited there is something else OK so let me we discussed this quick clean yes maybe let me discuss it so it in the paper we belive we did it fall electromagnetism and gravity that is due immediately got it is because I have the 15 minutes left and so that they would decide there were more about the the U.S. has yet to follow along a following up the 1 with the Arabs inside said well this site when you differentiate based the maximum power is 1 of our art to the Empress slump and the minimum is when you differentiating time this but indeed that this is why this thing is a wave because it has a component when our goes to infinity the leading terms the term which today is more slowly is the 1 over after the 1 our press 1 is the 1 and it is fast and and therefore at infinity this is 1 of our way which contains the and standard Haverty love affair and this is in what's in the old days .period Garion long called the acceleration with the it's the way which is 1 of our which had its it's the wave which budgets so this is the wave of also it's not only image abroad and in the static sense OK it is on this is that it is logical now when you apply this to gravity so you have to use Einstein's equations 13 arise Einstein's equations if you use these equations and as I explained before if you will a two-on-one engaged remind you that I define bound you knew his it's communion minus one-half of each victim you there when the gage is the new of its value and you put your hand in these games understands the questions if I write it in terms offered by the new is not aware of it but you know is my 16 g over the faulty OK so you have exactly the separated with the creation resources so I can apply this formula where a major today but if I do that one-legged again that page Marcio as as sources guarantees you 0 0 so I can write this formula for now let me let me sketch thinks so age because you I would like some of Of the at all something which would be like F and the miners over our where these FNL is obtained by this formula We've using you'll also replacing the source as OK at the end of the session will be given as a function of the sauce made this formula where as sisters your OK so it looks like something wave generated by 2 0 0 but if I do the same for Malawi fades I do its 1st naively think they're just putting this is the this as they are they are wonderfully OK it's just a number so I can the same thing I have again the formula where I have a but now the object which is the will alien diseases like that but there would be an exciting the Exide coming from the source because the source s now with the just I haven't index which goes over well so I will have something which contained 1 index and and as a function of T minus see you soon united by 10 I'd do it again for each IG I have a son but no idea of the sources TAG have to induces but therefore I have the same formula but I have H I J the minister of receipt of the OK so you could say I'm OK so I have expressed the solution Western integration in the summer of 2 boards that look like lamentable but there are not many people's because the definition of fermented bosses in the reducible representation of the mortician but now what is irreducible as I said is a signatory traced pretenses we've adding consider this subject GI GI and now in quantum mechanics books you would say I mean as a dancer and this is the tensile product call for victory and as the element of a space times something which is a signatory constantly failing this is OK case but that's about it other than indicated in quantum mechanics term of in mathematical terms this belongs to the representation the blind fencer Indiana there was a usable transformation but no I make a principle that it's not about so I need to decompose this thing in irreducible tents but technically what does it mean it means simply I have this subject and I want to as a sum of building blocks made of symmetry transportation and this is easy to do by hand because for instance you can write that this is process I think the Senator Frist people objection so it's part of the thing and then the wastes would be what the West you find that can be expressed In terms a what adding very intense please I have to invent and sales the I of the review should be done and Suriname encouraging matrix so you build from these objects invariant building blocks that you can make with these objects and you symmetry transferring them and then you equals the object so at the end you can lights adult of the formula with inhibitor use still here you make an intermediate objectives which I I should have said that in quantum mechanics and we know that this thing please there 1 plus DNA because the minus 1 what does it mean in insanitary transfer the usual formula for the spin additional became and and therefore what it means is that the subject which is not irreducible should be the sum of tenser which has and press 1 unit usable so it should be used to treat with Queen and 1 this as he should contain the part which is made of a pencil signature twist relief adding This is an apart with and minus 1 misses the the object with and minus 1 in this sense it is very easy to to to guests it is obtained by taking the trace I have an object here which are and this is here and 1 indexes if I make a ties which means I take the guy I leave and minus 1 in this is life can and that 1 of the ending this society according to his indictment the sum of our eyes so if I take a Tice between you and like if I have a G G K for instance OK uh if I think the trace G S S K I have effect of which is if I have 1 more index served as the sole symmetry free with respect to these emissions the seas against a meticulously with respect to the in this is that have not taste so this operation immediately gives me an object which belongs to the space can now to get an object which belongs to the intermediate space I need to use the exceeded because the excellent new contracts things which you have seen this as you contact to finally you of any in this is that you started with and therefore there is an explicit formula that I don't fight here which is a contraction between July subdivisions here like you you have maybe you take a here here have a minus 1 you have an eye and then decide you quoted idea again and you'd think symmetry to regard this and this and that but that soon fault and then the last object to this 1 which is an objective just and minus 1 and you put that a JI 1 2 and then uh and then this thing has finally address 1 this is like these guy became so it's just a simple and but this upstarts think is very concretely realized the this way and that we write the final result the final result you I have 5 minutes to write the final which is what I wanted to do the truth is that at the end of soul I had many multiples of so number when I had the scanner field the multiple expansion contained an infinite sequences of multiple moments like there was a motherboard pulled probable but just 1 dipole 1 probable case I had to sequence the skeleton like that of the objective here I have a little bowl in view I have died or I have the problem board coming off the air time here I have again GTI which is a very good idea I J so I have many objects actually most of these objects you can gage aware that this is now you remember about Einstein's equations these equations that are stealing variant on the continent transformation if I the danger which means I knew it was you look at that what is the general solution of that tax I you it is again motive or expansion so I can use I can decompose the general solution of this received gives me for all types of much importance because been the exterior wonderful sold this way I can supply 2nd gage away a lot of these multiples and at the end you find that the general solution of Einstein's aggression can be returned in terms only of 2 types of multiples which are of the so-called mass multiples and the spin would support and let me finish by writing the explicit formula so age about 1 0 0 0 years G overlooks the square some of the Bukit is just so that in the Newtonian approximations without the usual multiples of the sausages and Romanization convention it's the prove 12 idea of you over on where you really a C-minus obviously well know this capital letters you fumed .period plus them so plus the 2 size in desirable it's 0 0 general gage their which has itself multiple expansion stage because your eyes as minus fall the geologist secured some of miners and over factorial and D and minus 1 so it's a bit ability with and minus times at the time derivative of this disability boss of this multiple appeals not differentiated in time and here it appears With adding this is this way divided by then there is the next poor which appears secure some over my nose and then divided by this 1 factory an excellent idea is a minus 1 James the at minus 1 of you Over the white and then I would close gage there because diesel sigh I was size 0 and then H I J there is a "quotation mark tool trust forgery receive full of sound mind as an order of Victorian and the and minus 2 2nd activity update this aboard the same object wave of and minus to put you over on and then there is a term read it's a jewel the fall some of understanding of tempest 1 fact volume and the fearful that the approach of the growth from another just that the whites when epsilon AB I Jr j . derivative the and minus 2 to cover all OK which means that I have expressed the general solution of denies shiny Gration outside the also in terms of 2 sequences of multiple moments I N and Gerald Wilkins and and then you have explicit formulas that I will never have right here for these multiple moment as a function of the source said this was the main result of this paper with and you will be colonized what that means so anyway it's a theorem itself is just a result hired you it is the Ironman is an integral Paula space over this the sauce tasting table represented because there's always this entirely of their own and then he had you have something that may define the it's convenient to define the 17 0 0 plus says all this is great and then Sigma I to 0 I just the notation the sum of 2 0 0 0 and the trains because the main term here you find is that which is the time of Z the apparent defined here as X and which means multiple moments of what source of the source signal have taken at the same argument that appeal the argument as to be shifted by Z so I indicate this by today but this means this formula so this believes gives the usual you Tony and magical moments when I neglect 1 of seat because there I can expand this anonymous but I have a lot to answer for all times to and for the square meter 1 the 1st 1 but 1st To address sigh yes number 2 and this topped with index press 1 it's a function of Z X 8 2 5 derivative also the 2 0 0 1st 2 To 1 will receive full at this 1 and opposed to 2 of his 5 that that interest X AB it's a good time derivative of TB can cells and then I would not be right before the thing that you have a similar formula for general view accept and would like just the 1st this has the effect of just filled over the 1st tournament is this time Texas and minus 1 Epsilon Pi it's a signal the incorrect minus a term with the start was 1 look at the time there were 2 of best so these I explicit form alliances which generally if you take something which is static which does not depend on time this formalizes the people use the usual magical moment uh what happens here is that here you recognize what is this thing this season the cost products actually so it is ex 0 I as a vector OK but he's a Hawaii is the uh is the momentum density and locally so this thing is X costing it's it's the amount of momentum in the in the smaller volume so this is the angular momentum density so this objects is a multiple moment of the angular momentum density that's why these objects Jr if I take only 1 index this quantity is the total Raghuram momentum of the source if I take mowing this as it is a magical moment associated to the current of momentum so that's why scorers being moments in the sauce hot dates I have moments of this type in the source does not what they have already must type moments look at stop here uh so this is the the basis for being able to computing unimaginatively eating but now I was in the know as gravity but this is the starting point case of the waiver I've written the explicit formula which gives the decomposition in United Civil Wars lactation facility the all the general wave emitted by the source you denies thank you .period attention Over what is that that this is a game yes so I mean nominee gage here this is a nominee dish transformation that then can gage away and then you go in what is called Kennedy Court nominee gage where the the metric is only expressed in terms of 2 types of magical moments last speed instead no no no this all of these things are of here in the technical thing to compare the delta and by going changing the gage know these are gauging their objective is to these notable moments I can only uniquely fix they express the physical the bodies of freedom which I had eaten for instance later that we will find that the amount of energy and angular momentum emitted by the system is a function of these 2 types of multiple moments they physical things they generalize the lowest although product or forming a lunchtime was an said at a loss approximation the wave of probable objects and the energy flux is a function of this quadrupled here we are saying the wave is decomposed in the mosque whether Paul muscle to etc. and also speak well of speaking hoped to board and the energy loss is the sum of squared off the liberty of the mass to Poland must move gullible spin said their physical objects have