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3/4 Gravitational Waves and Binary Systems

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Bin hand the look a and the and
would case you I'm not in the end I'm mainly discussed what is a gravitational wave in what is the analytical theory of the generation of gravitational waves by general resources so
I won't come back to that within the mixture of postman costs skin and post Newtonian theory but at the end everything is expressed in important to expansions in the 1st part of the electorate to their will discuss the dynamics of Compaq binaries compartments objects which are a small radios compared to but the I think
about then and I will explain why the post Newtonian theory here are not fully would is satisfactory women they are necessary ingredients but they know and they lose their validity and the most important moments which is for the song signal of the merger of 2 black calls and for this 1 has developed his summation methods which takes perturbation expansions and are allowed to predict things when the expansion is no longer valid and in particular is 1 of the combination of was emission method which is the effective 1 but the method which was introduced he authorized shares in 1999 and 2000 and this led in work we find a sum of 1 2 the 1st lady prediction of the signal of coalescence of 2 emerging bloc called complete signals -minus infinity in time to press infinity in time this prediction as we receive is already very close to what has been observed but it has been in polls when of 5 years later in America where activities so numerical already started only in 2005 this was in 2000 but numerical relativity have tool in polls each ingredient there like the dynamics of the generation of additional waves and this led to the Numico attributing pulled version of the 51 but the templates for gravitational waves signals which have been developed in pilot he attaches we've assembled by God and in the states in the global fight some of what and in the last hour of where I will use think transparency is to be able to show nice pictures are in the show the comparison between the predictions from you and our European Union and the debate now there is a there is a notation that maybe you can get used to immediately that I will use all of our we would be talking about the binary system of 2 losses and 1 and 2 and it is convenient to denote the sum of the 2 masses by capital and and the so-called reduced mass of Newtonian mechanics which is the point of of the masses divided by the sum you according to to the which is conditional invitation and then in a paper with Natalie in 1989 it is 6 we ads in this paper many formula has where there were where fewer numbers like C minus the spatial Nuova and which is a dimension less number of the ratio of the affected must by that by reduced must buy the total mass and after a while we realize that it would be good to it would use the invitation for this dimension ratio and as the next great letter in the alphabet was Newell we said let's call this dimension as ratio which is finally the politics of the 2 months of the the by the square of the sum of the 2 months of this is the mention this number which is which goes from 0 to 1 4th in the equal-mass case it is equal to one-fourth as you can use find and that if 1 must is much smaller than the other 1 it is a very small number care and so it's interpolate between the extremists ratio made and equal muscling its way this parameter is equal to 1 4 we will also need later of this important conversion that given the sort of man given Newton's constant and the velocity of lights the combination G time so long as the either but the cube of the velocity of light is time which is a call to approximately 5 microseconds and more precisely 4 . 9 2 5 5 microseconds and more precisely it is known to 10 digits but we will not need 10 digits to today and maybe in the future ,comma additional experiments we need within digits although I don't know the 1st point is the dynamics of strongly said gravitating body the traditional perturbation theory in general treaties to say OK Einstein's equations which everything received last time of nonlinear disease for military the since lunchtime who introduced the strength of the united thing actually uh 112 you can do perturbation theory saying the metric is equal to the Minkowski metric uh plus a perturbation coal-rich new New and when this HE smaller you can hope to solve Einstein equations but the point is that if you consider frivolity we facilitate mass and it's important it is to able to finesse and innovative and if you solve this precise observations you find that it's say 1 company 1 of the components the time time
components but the other 1 of similar issue evaluated at at the radius of the objective of this thing is TO GET ME provided by all the distance from this but particular evaluated these at party for the pages you have now you need an OK if the subject is a neutron star a typical mosque in there is 1 . 5 times the sort of no d there is also an analog of this that if you take G times and undivided basis square this is now a 9th instead of being the time and this length is like something like 1 1 . 5 its secular Mitterrand who use while numbers so if you take a neutron star with wages of of thinking materials although it's not well known so you compute these 2 times the 1 . 5 divided by OK 1 . 5 kilometers claims 1 1 5 finally we factors do you find that it should be with at the surface it is equal to 0 . 45 In other words the military has I'd like to the majority of the votes -minus 1 I'm using the money it was the old 1 4 5 so instead of being a small deviations you see that the federation away me costly is enormous and it's half of them and if you think about all by definition a black hole suffers from lack such that I want to GM because the surface the black hole is such that it was Israel at the surface of a black hole activation theory is even more warnings than from a neutral so with the usual Whitfield mentioned is unjustified and therefore 1 needs the new porch to be able to deal with an important and with the motion of 2 black to neutron stars and an and there exists such a marathon this method that is what can be called the match mother-to-child this method was 1st using a sense by man in 1963 then all would like all the people to the at 7 5 and the rest of the nation by 1 the unity and then by myself all what we do it's just unity to for a while 1 of the world's even in the light Schleck off 19 82 published in the in the book edited by attending their relative to the appeal so what is the idea the idea is that you cannot use a its system of this type where you have this deviation so when you have 2 bodies like to block all of this is now a space-time picture of this is a space picture OK now time goes vertically spaced schools always horizontally this season the world line and this is motion in space-time itself 1 black or white were accused of another black coal under the neutron star and the idea is that you should use so according system today the is linked to the mathematical definition of money for the manifold is defined by that class of of lots of charts OK and then you just have to say that the at each object you should have a called system so the object KBO recalling its suspended as you call it takes a on file and the same thing for the father the of current the year Internet and and then between an outsider the 2 objects you use according at ex-New of this type and then when you and in this call the matchup is the global coordinate shops where you exclude what happened the other 2 bodies here you can use an extension of this died at the new procedure news at all there 1 of its new new at all the food you can use an expansion because if you stay far away from the 2 objective which and then you can use an expansion of this type but then when you are close to the objects to use a different expansions you don't use it all said that the metric is equal to flat space press
more preparation you said that the military here and you use coordinate system lets say it's convenient to use in distance from the 1st part of the Greek alphabet song for the time being as a function of capital X is equal To coffin undisturbed object that you talk about I can understand black or all an undisturbed neutron so you have the full of money now solution of Einstein's equations of you know for neutron the Tolman open of our cultural re-creations for instance all 4 dishwashers solution if you over auditing records so you put this as a zeal for all their own solution which is 0 the effects of care and then you look for first-order perturbation of these but the small parameter is something links with the where presidents of the companion OK so the small parameter will be the mass of the companion divided by the distance where between the 2 objects which is a small parameter because when you sit on the neutron star essentially the other companion was very you would be an isolated mutants also there must exist a perturbation theory of this type and then and then use matching met some of the matching of asymptotic expansions you have 2 different expansions to different perturbation theory 1 here 1 here and then you look for intermediate zone which is some overlap between this new law will coordinate system and the white 1 where the 2 expansions would be valued at and then you match the 2 things techniques for doing that and using this you can propagate information on what is the source if it is a black or a neutron star Indian External the ended and then this allows to compute and we think we want the metric and emotion of some somewhere line defined by this objective when doing that you can pull something which I had called the defacement off into almost culture which was the following things that you can distinguish whether you have a neutron stars or black or by introducing something which is the so-called love number off after neutron OK to actually to govern Bell's love number of a neutron star which I explicitly introduced in 19 it into uh is when you look at the preparation of a neutral solid external field is that the the coefficient which measures the fact that the neutron star would be deformed and that that the stiff deformed because there is a companion so it's a tide of the formation and this the formation will induce a quadruple moments in this new signed this quotable moments Wheeler of the field would generate a few by the way I remember about after introducing a minimum number of love is the name of an English scientists who had introduced love numbers for the title the formations of ordinary bodies but I remember when I gave a talk in the the also think showing bench last uh Dimitrios Christodoulou was there and then he said that go I thought you had introduced love number because this is the only on almost I said no no hope nothing to do with me but it is and so what what you find is that for a neutron star K 2 is Nunzio so it distinguishes a neutron star and a Blackwater and you find the effects here in this In this game so it's the effect of the formation of the staff by the presence of this companion gives a change in the matrix so it's this culture of dependency change of the matrix which is pop-culture to Kato there's a numerical coefficient then there is it's actually do in loss approximation something which is the title of the title role of field of the companion the 2nd delivered 2 of the Newtonian potential and then this creates a quadruple moments which therefore generates quadruple field uh here due to the formation of these times numerical coefficient which is actually the it's which is the radio 2 deceased power but you can only widen registered the 5th by our justice when the Treaty of
identity but for neutron stars or black coal by definition as we said the registers of all the of GM overseas square this season was 1 of 4 neutron stars and it is 1 for black coal and therefore formerly this effect when you say that this is a 1 of these Of all the you know 1 overseas square to defy quality square to the 5 all upon receipt the day and is something which when you compute the equations of motion yes because our proposal 1 of us is clear here is called the 5 PM approximation because this is something which corresponds to a term integrations of motion which is the overseas to attend smaller than the leading Newtonian older OK and this is very small even as of today OK so small that it it's too small to be computed to the into complicated so the idea was that this is negligible and therefore you're up to the that this was obtained within yes this was obtained within aura to the calculations this was what was new you can if you forget everything about lunchtime you can also find it's it within the Newtonian approximation of the passion that here yes yes but to justify its 1 needed this game but but at the end of meeting the scaling dimensional images look at the losers the said loan loses his years there is and this is the "quotation mark only at the index to heal means for people to work there is a fuller theory of writing it's a glove numbers that we have developed with her a sample my guys is sometimes but this was the beginning now that's what I want to say is that once you understand that this is saying that although we are talking about the motion of extolled the bodies neutron stars or black holes so a priori should be very complicated I have at each moment to take into account the fact that the surface and what happens to the surface once you realize that this body there is Donald culture is a very very small effect if you can't neglect this thing then you say the bodies should be equivalent to point masses can skirt tonight then seen from outside by .period masses because they have no structure and therefore the conclusion is that 1 should have a scheme where you can discriminate candidates to say scrutinize extended bodies like blackboards in their compact extended bodies and their fault the but that animates where you have only to World lines OK which means and action and then and then you look but familiar with such new question to that is to say you can forget about the scheme of the internal structure and say I will only solve this problem where the action is focus of these the interval of the understand their actions are we've questions 16 . G and then decisions In takeoff of space-time fruit of GE and then you use critical nicely with the quiz and the bodies by World War II .period masses which means that you put in action which is miners in of the it said the new this is the new can you have world lines and around the world lines you rights the metric the of the popular time then this is the action and then you solve this anomaly called in it yeah this later this year it is dealt with by an extension of the rich which is much more complicated the trees being but it can be done about and it is I be is developed with the spin through very high older I want to give them and did so now what kind of the creation of so at the end India's Einstein's equations for instance on a new vehicles in Beijing over the 4 some of the other 2 of this place energy-intensive of northern miners and taste around in mentioned the indeed where this thing this is the valuation or the
action with respect to Jinyu so distressed energy transfer of .period last year an integral and this is where the world of the pop-up time along the world lounges including using the new using new the incident the functions of a minister but now we have a problem because you have that stuff functions but by way to complete last time I explain how you define gravitational waves by solving Ashland creations and producer by using so-called Monday called it's convenient also although we see that there are problems with this to use on Monday coordinates in Genoa Science on Monday called in it's all defined by the condition that the pitiful of symbols contracted the disease fuel which is equivalent of saying what you sign that's a demographic nitrate which enforces grouped under the command of the report was you and by the way the Palestine or widely used this condition in 1912 focus some people say let's call it good don't go there and stand in 1912 knew about commonly called and worked for years trying to reduce the V-chip and using them money according and then use them for gravitational waves also in 1960 when you do that the advantage that eventually answer here becomes a diagonal route to the operator in the sense that this if I put nose 2 then this gives Ejiofor bitter the effect of the Virginia new pastoralists of dingy dingy 1st Liberty of quality contracted with the universe metric a hearing on management to this is done so yet on using this contracted in some in all possible ways exist so so the advantages you see disappeared terror becomes so broke operator Bettina coast this time but when you solve this equation by saying I will look for a solution on the pageant union called a Kenyan presage new this appeared to have become the Mandelbrot flat space-time and then you can use perturbation theory saying that the top of the question you have at Lowestoft there for instance is not on their community will at 1st told there is minus 16 apology over the phone 2 million you can't trust the miners well over the last 2 and that the new knew he was we have to Nunavut 1st is this quantity in which you replace the metric by it and you knew of and then this is fine this is where the final but now let us see what it means 1st I am computing the match .period clinics have to Werlein .period extra fuel .period this thing I can say I cannot at present by a diagram saying that each new at output x this is a call to the inverse of this approach which is the Green function for the tangled on immigration so itself so this of this thing and then OK so this week the line means at their new news story that annual flower at the Nunavut government has slowed over the mind to continue and to have fun without getting a pledge action Alfredo fault investors times due to Green function which genes which satisfy the the delta function of X minus 1 which is explicitly known in 4 space-time dimensions you think that we call the Green function so you can solved by successive approximations this diagram adjusting the seasoning arise brevity I and is not here means I have to new new 1 this team you knew which is which is overweight done by you delta function integrated of this world so the sources is something which vanishes everywhere in spacetime except on 1 or 2 world lines and military can be returned at all the this express the other than that which is this so this is the defense generated by must say this is the thing generated by must be at the next approximation when you solve no this accretion to the next books initially of something like that number of rich new Seguin approximation in the right hand side contains various types of terms like in each language to the agreement involving 1 due to To this day OK put like inside plus all those stones which shall from any politicking the 1st activities plus the term which is which comes from the source if you expand the cells the cells thereby infringing you knew him here so if you expand your also H 1 finds the and now this gives Exxon diagrams in perturbation theory these diagrams hold this time I computer .period X of something generated by must reminds me and then they combine at this point and this CU vertex means that time solving outstanding creations with the quality of their U.S. relations With all just all over this but that the nicest where yes I know that the explained this now if you relax Einstein equations by Dominique condition you don't have accretions of motion you can solve and find questions for motion of the sources because you don't have constraints on the motion of the sources at the end once you have saw for the metric you want to impose that the Matrix satisfies these conditions your solution must satisfy this condition and writing the these imposes the constraints on the sources and you find that this constraint is equivalent to saying that this is the equations of motion of the world line in the metric in which they are bed OK that's what you you pull the former technical so but it's convenient to solve the thing without imposing the questions motion and then why the acquittals of motion and then at the end use of gracious when everything is satisfied the point I want to make clear is that there are not only desire Gambardella diagrams of this type OK yeah and now sorry politics and now so if you so so yeah backgrounds which I like firemen diagrams could actually those diagrams equivalent to what Fokker
flush you and the people in the 1920 s and 1960 introduced before finance and firemen I had studied the papers of this people and then he understood that even then it was convenient to the wall diagram uh so fine and was interested in quantum problem here we are interested in the classical problems Raja saying here that perturbation theory is perturbation to we have Green's function concatenation of eroticism and function before the full polls that you have indeed the firemen was for this was given in a paper by me and political fires in 96 so it's communities stood it's a language OK but what is not language is that you see here think switched from the quantum field theory you would call a loop you can look is a diagonal a topological free of this thing now what they do mean groups classical groups exist OK and and that means set for energy effect if you compute for instance what means this term OK in lowest approximation terms mean that I must computers like the universe that ambition which is always supports mission in the universe that class and all of those in the gradient of the 1st metric each 1 but the 1st although metric is an enemy of our OK is is the good news the group potential 1 over on the Newtonian potential if I'd taken the liberty if I have any overall Square if I take the square I have in the square over fall and now so this thing in this article were stalled but this is an interval in space the 3 X with the ex-field-marshal this annex where I am 1 of them square over thanks to the fall but this interval is divergent because in the space dimension and I have a function which is singular like 1 over the distance to the 4th 2 point in space OK so this thing is going thing and and this thing is what this thing means that I have a car action to the source this time X minus sir they and then this multiplies in space it becomes this and this season they may divided by the Newton potential X minus Z by but this is the that function is saying that I should compute thinks X accords that OK but at that accords said this is infringing focus of this thing is also in the so so here we have a price to pay and for was very convenient to say I mean that there was a poll that you can't skeleton Isaacson the bodies by .period masses but once you do that in the nonlinear theories like Einstein's theory you have infinity to cope with the pain and I will before discussing how you deal with that let me say that he I was discussing how you compute the the field once you compute the field you can compute the equations of motion and you can also pulled over and so that the days of reduced action which is called in the In the gravity literature for action because it focus was the first one to do that for electromagnetism OK and for electromagnetism you have only 1 diagram the pictures this 1 which is the 1 foot on exchange between 2 charges and Foca in 1920 wallets and outreach washing in 19 15 already knew that and Fokker wilting 1920 What is the actions of exchange of 1 of what we call 1 know between 2 world lines which is a classical opportunistic version of the quarrel in the interaction you can do the same thing in gravity but ingratitude the action of 2 world lines as the 1 guy meet extension well now in a sense you you close this thing by putting the you compute the interaction energy of the 2 will blind I don't want to let me 2 years ago I have given the full set of lectures on on this the Fokker action so they are videotapes of this so people interested can see on achieves website the full discussions in French but the blackballed drawings of a new language uh so you have that you have now leave group effects and then at attire although there you have things like that but you have also things like that OK you have things like that I don't think so that's like that uh and the with Natalie we computed this thing up to 2 groups of friends so we will give them the results null but what I want to discuss know is how you deal with so this means as it is clear here and is clearly at me and said it effects it means that when I've appointments this disappointments generates a one-over off-field this 1 over a field as an energy is an infinite energy concentrated that articles you and it is spots in gravity any type of energy gravitates so this has to be part of the business you cannot GET completely without knowing what it does OK so how you do that good may further the 1st good method which was used because when you do that at the for instance in the Book of land ownership where this is treated at 1 PM EDT this is easy enough that you can say OK at the start of the new and let's not worry about it when you at to PM this is more complicated there are things that divergent says that depend on external viable so you cannot just pull them and say This is the old you have to do and a convenient way that he is undated continuation which in the blander introduced by month's end and I won't know to explain this quickly uh and by the way this fall as the early methods using quantum field theory the leader in the 49 and thing like that this was introduced by Marsalis and was used false on wheels In quantum field theory that me explain this my further which is the 1 I had using 1982 to the figure relies on the bench with things that happened at the segment posted to an approximation and the basic
idea of mustard we is that the the polymer of deployment comes from what the prime comes from the work that you have point particles which therefore that the function on world black and and the idea was that the place the delta function by something which is traded call in some sense and and then do the computation with this arise that the functions and their plagiarized version of the delta functions India mention is what must is called a Z with an index it in dimension and it is the following objects it please minors the technical the square inch of all 2 of the Power 8 miners over to divided by some functional race that I will not defined explicitly times step function step function is this know 0 and 1 after steel times a 2nd step function this two-step functions means that this thing is Nunzio only the future right corner of X OK so it's 1 here it's 0 all-time but essentially this thing is is the interval square this distance square with a minus sign to some power but but the trade of Monsivais was that this thing satisfy the the following identity was too and I said Please convoluted with b he then it must be OK remember that there yet said a word he thought it would have been the sorry this this fall here there you go sorry this is on but not the way predator acting on their it to is minus Z impressed and so what is a day is a complex number OK and in a complex number H here is a politics of government functions it's a numerical coefficient which is such that when you do the convolution all these things in spacetime OK it is somewhat complicated integral you you fall back on this with the the property that it's the same function but the index a sleepless by the sum of 2 things now as you can see immediately Z 0 From this thing is unique for the convolution a date that the all is that they OK so it is the unique convolution thing but what is the unique on pollution into the delta function OK so sold for convolution in spacetime OK and mention so that their role is the delta function so the idea of waste is you he place there is doubt which is that the old buys that they wear a zips along with a smaller number and then you are going to take this small number 2 0 but instead of saying it is smaller you say I do NOT continuation I take it in the complex playing however complex airplanes and then I'll be placed in Einstein's equation the side that that that function by this and then I do the computation fall in the value of rates and then when you do that you don't have the single writings you can do all computations when a is is in a certain region of the men of the complex place why is it so because if you look at what happens fall below the three-dimensional and indeed all of the world line it was a point particle in space was like that of function this is replaced by 8 times the distance so I have a point in space OK them and then I that function is something which is only here now I have the to at the political of the distance between X and Z upon the release thing is of this type In 1st occupation it is this when it goes to 0 old this goes was he also he goes to 0 except that there is a single item like 1 of rock you and then you find that this thing is indeed a representation of the that of function when it is small but the idea and is now the that we're in Phoenix for instance was the reportedly Freites and and when you now what is the equivalent potential the Newtonian potential is that minus 1 of these but this is a lot less of minus 1 of these and then you find that this is up to the power a minus 1 that is to say a what before was 1 of now becomes to a different power and when I compute for instance this finds the gradient of the square I have something which it is a lot to do to is minus In the middle of a space but this thing is the higher value the real part of it is large enough OK so when you are somewhere here this is a convergent interval and this convergent interval is also inundated functionally and then what you use is the poverty of the continuation of the nite a continuation of identities and finds that the basic idea of waste is that given any function have defied compute disentangle the ought plus infinity thanks to the game before that date the according think loss have finally when you use for a coordinates they will be all despite what was before diverging there will be something like that where it is able to remind us for something a combination of reasons some integer but this thing the 1st decomposing in the interval from zeal to some number of interval from the compressed Infiniti you put into all into now you look at each part of the problem here is that this was diverging at x equaled 0 but if you take that we'll part of the loud enough you can integrate the spectacle and then when you do that you find so that the convergence here and that it is a mail Norfolk function in the complex plane that is to say you can extend its uniquely in the form complex play it but the pies you have to pay is maybe undervalue which is a good deal maybe we'll have a pulled OK if you have a board you have to do with the police have no this is a way of defining the thing that and for instance and if you compute this way this interval this same is when useful role coordinator of a fall by
then I have the honor of square I got to the crew a minus OK but this interval can whites as interval from the will tour of the old 4 of the other 2 to A-minus to 1st from of the world to infinity of India on Of to A-minus to each 1 of these inventories convergence in the parts of the complex plane OK if you compute this to intercourse and if you add them you will find that this view surprising the talk that end and same thing with the delta function then becomes you so at the 2nd posted on an approximation now let me summarize work at the 2nd posted approximation which means when you go up to 2 OK it was shown that the unedited continuation where it landed contribution goes to 0 the use of finite results so this is a way to computes something new computer metric you compute Grecian solution immediately at the end you have to pull out that what you computers what you want you have to pull over Her which is important in my lectures of his that it is a social ensuring aggressions and that it satisfies all what you want and that it satisfies actually the polls was to say that this season this concept solution of Einstein's equations as a function of 2 World lines OK with some of the culture of other world lines that the were blindsided looks like a sure solution only now Shoshone solution because you knew you find this way older expansion of the Foshan solution to 2 PM case during the period for this and then help Poland that if they exist a solution Ranch a question of this type which looks like trusted in the technical sense will let the team equal To the unique to this solution we replicate above before pocket so that was supposed and that's because this was giving the correct equations of motion now what was the result the result was that the 2 begin dynamics and and 2 . 5 with that that the uh so I've written here that the new to the post expansion of the UN is Tawnya OK here we were the nominee called in it's actually there are subtleties numbing coordinates that does not exist and June and there exists a general rise of Catskill 1 genuine and you can go to a different schedule coordinates so that your ordinary until all that has been discussed at length I don't want to discuss it again and at the Newtonian approximation of 2 bodies at these 2 square over to minus 1 of our key I am using scale of things whether GM of is the ideas is divided by GM and the new also delivered to the master is also scaled the way to have something that's OK otherwise it's makes and and will be sold this is Utah QinetiQ Durham minus potential energy minus 1 over at the wanting an approximation which was were 1st worked out by thoughts in 1979 and staying firm of mine 1938 the Lagrangian and was 1st written down by fixed and holds in 1950 you have this which is already you know there is a key for performance by the way I I mean vector of his square square accepted that default you have to square 1 over on you over and is the unit vector between the 2 bodies that were talking about here you have ended the which is the ideal moment to square and then the other woman wrote to ask where interactions because this is a you just have to be him if you want to the you know at 1 PM the 1 PM there is an ordinary like Bollinger as discussed in on the ownership of the surprise with Nathalia remember was that they were a few I'm saying that does not exist ,comma ganja OK and then in the when we found the equations of motion we looked for London and we found that there exists the demagogic but he had to depend on the higher derivatives at 2 PM on Wednesday at 2 PM but you can also eliminate at any given all the actually you can also eliminate the idea that would use and an ordinary London and then at 2 PM and this was the answer we got so you say it starts getting no missing OK Malta and there are actually 11 different terms no motive and uh we'll get the counting and you see that the coefficients starts depending also offer Of the question new which is symmetry must facial so a number of new issues and new over and goes between 0 and 1 4th and you have to MCI terms quality can you uh at this level OK now at other times in 1982 In the polls that at 2 PM this was fine I have also realized and publish the fact that there is a pool at OK that is to say I think in the next although they will be Coppola 1 over and then to me is too surprising because this fall for saying it easily Paul blames all the fact that the point particles some look points but expanded objects this should be at 5 PM not steeped in the solution of the the thing came when 1 explicitly computed the thinking fit but in order to compute the Fijian thing which was 1st done by Nancy Shifrin myself in in 2000 1 needed a better legalization methods because actually uh the two-goal 2 to 3 PM and you need a refinement on this thing but this refinement was invented after was invented in 1972 to independently by Bellini John and by half and investment man and this is the dimensional legalization of dimensional regularization he is also the mind is also a method of realization based on an 80 continuation but instead of smoothing them at a function in an artificial way what you do it is In fact you look again at his former of this formula with I haven't done any space-time dimensions is OK and then you find that the critical thing is that there is a power of the isn't alone which is a mind and here we were using the fact that I do and 8 a continuation in the power because I am saying I kept the equal fall but I do it a continuation in it that mention a legalization says no that says the delta function is the so I put a cause but I play with the if I whites the Green function explicitly as a function of the and I take the dimension as a complex parameter now so if I'm able to compute the subdivision fearing weathervane function is at each stage computer than that mentioned the and I have an explicit analytic function of the I can now do and it's a continuation of the and this method is practical in classical gravity and works and it gives and then it generates a solution of Anthony questions which is what you will let me the was he said it was the latest in a long time you know what are what do you think you have to answer it's been that this let me explain this to explain that actually it is different because when I do a instance when I do this computation so when I do what was the in thing all of us Spain now it convenient to capital D is the space-time dimensions OK it's convenient to call the space dimensions smaller do this will you remember what is this time is the big 1 and the space is the smaller 1 so small that is the president were actually is a complex parameters when you have the table over a space of the derivative of each of the new potential Square OK what happens is that age is the solution of the of the I'm staying yeah I mean
you denigration in space dimension the solitude and of us more down to the minors to the Newtonian potential is linked to the dimensional space it's 1 over because the coffee but when it is not clear that the material also the which is here for this in thing glory is the job times up to the the minus-1 finds out to the so it's times are to the the minus 1 of those square for the 2 so uh but this does not cancel the to do so you can play with the so you get the same result as before for the delta function what happens is that you know however that the function but you compute on to the the to at quality but these large enough this is 0 simply because it's Tiger this thing so this thing is there in in that mentioned the focus at the at the CPN approximation that really feel in the sense that if you project the back of the diagrams here where in spacetime OK but actually the Green function which comes in the calculation is always the brain function linked to the publication space than the plus minus 1 and that 1 of the important diagram most of this type you can wear this thing means that I have sourced by the mass and 1 I have the feeling of of the lines so this means it is a term of older and 1 cube why he had nothing to do with this season of older and 2 square because of 2 lines and then this diagram means that he and I have a scan the for but which propagates which is the longitudinal type GArlington why here I have the gravity wave which the transverse transvestites less as explained got it obligating which is sold by the derivative of these OK so this is a this season and intervals and which is as you can see we leave creative because technically although these are classic it's a classical theory and in quantum field theory you would open the lines and not call them group .period comes when you compute them not in the 40 of them in but the next based there have really as difficult as group diagrams to compute and the number of routes is proportional to the difficulties OK now when doing the computation in Canadian accordingly which is unloaded there's no means never having informing them from general activity you find that the answer is finite after a continuation in Spain except that polls but the polls canceled so it's dedicated because when you have won over to the minus 1 to you want to be sure that the and then to to get the calculation white you really need the meringues and that this was a calculation that we've got Schiffer in 2000 where the might was essential there and then you see uh now let's quickly ended the result is much more complicated and also for the 1st time the output from the local numbers coming OK here about fashion on of on the way here you have the right to by square 6 that comment and that when you don't have calculation wouldn't stop for the break all 3 have essentially finished yet that he finished and we will have a 15 minutes of the break when you do the Friedkin calculation the nominee coordinates you find that in the days aboard so it looks strange because there is no border you do this in a certain coordinate system and the use of force if you do economic that which is sold this confirms what I had said in advance continued into the review board reasonable but the new information is that this sport can be normalized away in something which is the the position of the world lines because when you do the computation it's a computation which depends on the choice of a world in spacetime OK and this will outline the president's what represents the fictitious center of mass of the body but if you have a black call or to stand that does not exist the central OK it's not buckled itself and modified so it is something which is an intermediate technical object this wetlands and you can shift its you can say I have a better way of life I have a physical world blind in space-time which is like a bear where blind plus a shift of the world line which can be singular like 1 over the ministry and this is what you find that it is a few shift the Werlein mathematically by something singular but which is more because it is G he would receive 6 or something like that in all the pin all the it's a smaller term at the BP older and then it shifts away all of the reports there and then you find that the result is the same agreement to that OK so this calculation and Michael the Nets was 1st done by myself and Jim Esposito fires and the with the honesty and Schiffer we have shown the equivalence between the 2 so this is the let me mention briefly technical tools just because these calculations you know they can be done because I think they are intervals to computer OK uh so just to give you the flavor To do this computations you need some forgeries results Samuel mentioned here and the old results like the old result is the following is this diagram this diagram means that I have a must .period at points that 1 of us .period I bonds at 2 and then I computes the gravitational field in general activity due to the nonlinear interaction between the 2 I truly diagram means a lot plus minus 1 must win the support of plus minus 1 which means that this thing is not possible in the form of blown over excellent 1 explains that took effect this out have the Green function for the Laplacian in the space-time dimensions 1 all 1 overall for 2 points and then you compute again the press interval of this this Cinderella is already nontrivial to compute it was 1st computed by Fox in 1939 and actor it's a very simple answer and I remember when I was a kid I did not know at all to the work of folk and I discovered that collects resolved by using some so I like to read this thing is I want octopus someone to so it is the lot of the perimeter of the proponents of the of the point of the distance I want to use this distance 1 of 2 you take the perimeter of the triangle logo of this and satisfied this thing this is fork and then there is a very nice formula due to of which are ridiculous flights which is when you want to computing at the goal of this type which is you have to points in space you take the distance to the point where I want to some power you take the distance to the 2nd point to some power where these arbitrary complex numbers and then you can do this explicit computation this spiral 205 and then you have the the distance between the 2 points to the power of the testing procedure and then here of the part that I would only like to that of the gamma function 3 of the gunmen of any of the government I only write a few of them to indicate the torture so it is like a generalized beta function with with the and and the young mothers and their marginalization and so you need to this experience of intercourse to computers the thing to be able to compute the CPI and we will briefly discuss them at 4 PM in the world with the from the calculation at 4 PM has been done on the diamond buyers say honestly myself in 2000 for faulting OK there and there are about 1 100 terms in the middle indicates it's a very complicated thing and there is a subtlety which isn't known locally entire new thing but that's and discuss when needed OK let's have a 15 minutes
but while I mean raising that I just want to mention that there is something very interesting and subtle happening and the 4th Boston to approximation Ritchie's that up to this level 1 could approximate the interaction between the 2 world line this the interaction between 2 unmentionables goes Holy Father Green's functions that at the lowest although this with other couldn't function says that the gravitational interaction Popeye gets a diversity of life and between 2 close objects this is a small effect compared to that of the space the move in the meantime can but at the 4 PM approximation there is something which is nominal called in time the Green function contain something which is anything or over the infinite bus and formerly the dynamics depends on something which is a functional on the Infiniti bus and for the Conservatives dynamics of the infinite future also this is linked to something we had 1st found with little a 1988 which is a breakdown of the pinup approximation the notion of the ozone you gets completely I mean it's very subtle at fault here you need to take into account the Infiniti past In jittery effects of what happened in the sole civilian usable in the dynamics of the source now OK but this is a very small affected but it must be taken into account the explicitly OK now coalescing binary black hole but I wanted to use this for the environment and enhance children's shows that there only 1 the 2nd 1 you just don't know I've written surfers that 1 a little mind terms present oral called terms which is an integral of in have not written and oral culture here and then when you go to the center of mass murder things this reduces the number of terms I don't know the exact number of weight on only the first one in the last 1 here it it would need full blackballed toward this thing now so the motion of 2 we are now discussing the motion of 2 black holes OK yet it's more complicated than now they still recalls the go on each other uh because of that we found interaction there is doubt that forces in the system um which we which were 1st directly computed by Natalia and myself in 1982 so part of the Christians emotion as a dumping term which means that the 2 address gets closer and closer and then this means this system will go faster and faster and they made gravitational waves of increasing intensity and uh increasing frequency now 1 can distinguish various appeared so this is a space-time .period run time is minus infinity in about 4 fall for most of the motion of this to block codes in spacetime 1 can say that these inspire all is a diabetic or take ideas that he also had a good idea but it means that you can consider that the 2 objects at each moment on the Variable albeit and you take into account like a circuit of change all the parameters of this declaring albeit as a function of time as a thing as an idea about the change OK but uh when you get near manager and the former it becomes known that diabetic the things starts falling well can't uh so there is another argument that the relates in inspire becomes nonenzymatic then there is something we shall describe which is called the plunge and after the plunge theories the where did and then there OK now I'll do you compute this 1st there is a nice simplification that had explained last time which was followed by Peters in 1964 using the lowest all their emissions of angular momentum from the system and energy from so you you computers that use the quadruple approximation of lunchtime 1960 1980 in which gives that there is a "quotation mark away from the squad Apollo wave you compute the energy and sent to infinity and the angular momentum also all you use what we found was that Natalie which is directed to addition reaction force which takes away from the system in the energy and angular momentum in agreement with this squad of afford thinks and then the 1st thing you find that Putin's found is that the eccentricity decreases faster than the sum I major axis of the albeit so more precisely enroll with approximation of the eccentricities squared divided by the appears to 19 of material terms which tends to win I think this constant and therefore has the period gets shorter and shorter of the eccentricity gets smaller and smaller which means that the system even if it started interviewing really think uh on white albeit with become Suker on wide albeit before this why don't shrink to a smaller and smaller although it's a it's a weather fast process so it's improvise a lot because then you can discuss the inspire all along quasi secure albeit so you can do computations to buy all the impose Newtonian thing but assuming the 2 object's goal in circular orbits and this simplifies the competition OK now what these are sketched in the 1st lecture what is the logic of doing that the logic is that you need the equations of motion which includes both conservative effects and admitted that being effects so if you are in for instance in 1 of the no coordination in the conservative dynamics is written by an instant secretions for the relative motion also you eliminate you go to the center of mass and you eliminates the motion of the 2 objects and you replace it by the motion of the relative motion of X 1 minus 6 2 thank you go to the center of mass where he unaccustomed to the court to 0 and then in the center of mass you find that the action depends only on the hour we are pleased that his X 1 minus 6 2 is that related thing and added that is a few of the action in respect to our which is the momentum conjugates to derivative thing is to he won and minus the 2 simple so the 2 things are the old and the first one is the ability of momentum and this piece satisfies question promotion the usual and on equations of motion where the Estonian if you use in theory is given by the you don't approximation this 1 kid and is to appear and disappear then and now we know since last year the fall pin prick OK but if you if you like this the 2 objects to go on in orbit for a atonement OK they don't lose energy it's a conservative systems so they would never say we know that there is a additional action fast that you have to add How do you compute the addition reaction force so as I said in posted the H is the son the 4 h entities insist square the problem Muslims system plus each Tonya was 1 of us to square each 1 PM presented so you have an expansion of the dynamics in bowels of his square overseas square on 1 side now when you compute this new computer it's by or by computing the waves emitted at infinity by the myth of the brochure than Wyoming felt the five-week-old in the 1st lecture and this also which is quite complicated then you need to compute intercourse tentacles you can compute especially for Sitka bits where things simplified not what is too high posted to approximation CPN and the . 5 .period looked at 4 PM because there are problems there uh uh the uh the in the rosters of energy and angular momentum enacted a new computer essentially the loss of energy because it's enough for computing what you want full Bates so I don't fight again the formulas that I that the 1st thing coming from the there's for their uh OK and now the usual approximation that you find in many papers on this but which is not enough that's the point at which is to say OK I have a system which has some energy energy of the system
and I want to write that this system the losers energy so I really liked kind of balance equations saying that the roster the the change of the energy of the system is the court to minus the energy lost to Infiniti sold minus the flux so this afternoon the flux of energy and the forces of it inquiries I Albany but the system is described by only 1 degree of freedom you just need to know that radius at each moment all the morning valiantly the orbital frequency omega sold only got capital of that to fight over the period of the system and if I wear in Newtonian approximation I would say I would capers law which says that GM is on target to it this is the so-called John Weaver called C D 1 2 involved because you have been to the Our wonders public to and this would follow the adjusted to remember where only got 1st of all it is they can't and you remember it because the velocity and plenty of candidates use velocity associated to that you define as the usual formula only got antsy uh you know you can express if you want to buy using this formula this formula if you were using also the new turnip automation you have the energy to you over to the uh so you can express everything in terms of of the subject because you have this equation to eliminate aid in terms of from a guy you eliminate a intensively so you find that the energy of the system is minus also knew the square well use disease which looks familiar but there is a minus sign because there is the 1 that civilians fearing the minus 1 over all of the attraction energy is compensating press 1 half of the new square then the you have this thing if In these objects are function of the you can't say it's a differential equation in the final dysfunction this thing becomes a partial derivative of the respectively times all total derivatives times and adopt the view over the the gold miners had so I could say that this equation to mediocrity the year minus the flux divided by the number of so I can compute VI as a function of the government did not as a function of this is so if I know this too like fall 5 approximation I can compute this to some approximation and that these new differential equations to this approximation let's see what the gives at the just for orientation looking at at the worth of a Commission which is the political formula the flux these g over OK so as we said This is due over 5 0 C 2 2 5 this 5 is 2 times the speed press wonders why there is a 5 an uncharted I new it's implicitly then you have the severity of the probable moment square when you compute this fall to .period masters and get 32 over 5 g New Square that's a new plants oversee could attend OK now you can sort of overturned on the blackballed all the equations should that you can solve the as a function of time at the request of optimization and you find that you receive is 1 of 4 also called for constant integration constant reminders fell to the 188 Where have introduced style which is the time there is again a coefficient 150 followers and I explained there is new which is defined there and then this is a time in seconds I needed I mentioned this equation the this is a velocity in the usual units in the overseas appeal number OK uh therefore he I should have kill number so I need to divide the by something which makes dimension less than that and that this is a gem of a secure so anyway you define something proportional to time and then you get that the solutions from other the lower lost all their spying and portable formula is given by the which which means what which means as a function of time too there is a critical time to see where this thing is inching its way went to minus infinity the overseas tends to 0 and then if there was a single lighting which is a weak Nova -minus 128 so it takes a very long time for the velocity with his slow slow slow and at the end of atrocities in that sense the overseas should stop To infinity infinity at this stage just fall pedagogical ruled the reason also the independent or approximation of the the gravitational wave frequency only got his twice the orbital frequency reason is that as I explain a 1st lecture on the ABC-TV of the gotten used to and this is the city in other words there's a probable emissions means the frequency of that you observe is twice the orbital frequent OK and you can compute using this formula to their that only got a lot of Omega square so additional weight only is a frequency omega . is that as the dimension of a frequency squared the inverse time square so if I divide by Omega square it's a dimension this number this dimension this number when you compute things is a certain number 48 over 5 times a certain g ended only got wrote to secure to the 5 thousand now this subject here Is something with the dimension of mass and this subject is for the mask and it is according to a new which revenue is the Semitic must assure defined their times the total mass and so this is the so-called choking why is it called the Chapman because this is saying that at each moment the frequency of the additional wages increasing OK but I can't compute that if I compute this number the frequency don't always increases of David T. developers group frequency and I express it as the Pfeiffer power of the frequency itself I need to put in informed something referred dimensional so if I can observe this chirping if I can observe the value that only got you know if I have only got a function of time again and I computer at a certain point you Megawati's Omega adopts what is the slope I can compute the chop must because this is the 1st thing world could that could be done for this event you observe something it is chirping you can compute the mask from its uh and then in the case of this you find that this mass is something like 17 solar masses From the less jury would be better to define the chip must reform you here because then fold the systems before new would be essentially 1 because is 1 4 for the equal must assure and as it is the maximum it is equal to 1 or the maximum for many months ratios then we we know that the truck must is essentially the sum of the 2 lessons that it would be 65 source Modular Direct Shift cosmological achieved but now even if I am at this approximation and I said OK now let me go beyond what he said because of the year now she desires exactly this is unknown should now be 5 each of these functions FOB ease the lowest all their soul which is 32 to evoke 5 New New Square the to the 10 times collections now we are going to see in Powell's of university which contains both odd and even have something at the end of the day and the energy this is the Newtonian minus one-half of New Square Times also the overseas Square which is 1 PM collection the oversee fault which is to be a university 6 which the clear and started to fall 6 and no 1 even knows the overseas through the book and then would be asked I will use the best of formalized and I need them now let us see if freed 1st lady need then what is the
last time there was a question by you know saying What does it mean viewers see what is the the oversee at Melchor look at this the answer to this it depends on how you define the overseas OK but let us define the overseas observation by what he's uh yeah if I combine this formula he and he and I find that the is a call to N only to do 1 thing and the new turn automation but now I can define the given uh given a gravitational waves with frequency on I can define and associated orbital frequency which is on the government to it's a conventional think then I heard here defined something with dimensions of velocity to say that it does it without mention Uganda thereby here then it becomes Cuban inside one-third but now we know Jim Morrison Cuba's precisely time so this is indeed the mention and now we can ask as the from the effective 1 but the methods that I'm going to describe no 1 can compute in deformities and what is the gravitational frequency at manager in a precise sense OK now when you and we read all social and it could be our you do that now India before medicines the dietitian with frequency for the 2 to more than equal to an "quotation mark move at welcome progress secure you find that this is the 0 . 3 7 7 OK or what the viewer 150 now this means that the GM God formerly associated to this is half of these so it's something like 0 . 1 9 became 1 8 8 8 5 this yeah sorry it's so it's fine equal mascot but this is the case 1 event because the correction due to new on the digital in this case now but if I computes this is mountain now I can use this formula to compute the overseas at the merger .period equal-mass case is just a convention just to know where we are in the expansion we want to know what merger this expansion what are the terms that can and then you find that this is so you can do the calculation have this is the appointment in 1 thing and this is the 0 . 5 7 4 this is why 1 says you know that this is roughly half the velocity of it's even more than half the velocity of money that's manager but now he I have expansions OK but informed their coefficients locate the provisions on 1 working eye during diversions goal fast and we understand why the goal fast and if you can compute the term and the overseas 6 for instance the plan at the overseas 6 but the In the flux as a coefficient which is 1 of the 23 -minus 16 round of speed 2 the overseas fans this if now you put this number and you put this you find that a merger the the political action here is equal to 4 . 2 7 compared to 1 so you have an expansion which is 1 plus press 1 press to process the press 40 something which is not at all small compared to what so in that sense the overseas 1 of is not 1 of 4 OK it's more like 1 of the the terms in the expansion comparable to the lowest auditor for this shows that you cannot at all use force you to expansion at notions OK now thank you have to stop before Tuesday now also another way of saying that is that we here are the ratio of 2 expansions so when I have the ratio of 2 expansions 1 president square divided by 1 but I can do 2 things I can leave this issue as it is all I can we expand that we think all I can push it on the right hand side winners so I have many ways of solving this equation which we hadn't use with the idea and set up a cash recalled to 2 to 3 years to get the OK you you can do the same balance equation you can solve impose due to his theory in many ways and you can compare with other solution same prediction at the end and then you find that the divergent that's new manager of the predictions bolted to the front so it shows intervened secretly that you cannot a priori tossed posted expansions but 2 marriages occurred just before but there is worse than that witches the Commission's they contain the conservative forces and action for the I was saying the additional reaction falls collections due to I posited that become larger than the leading although there formally actually there are not we would see that when you some then not object that the the in a sense the Spaniard and credible thing is a good approximation but you need to something but the undertone is worse because for the foreign isn't he and I have I have this summer's you if I look at the Dusit could all be working how do I get sick at all bids from this from the 7th onion that I take a page and I said a circle beats is the effective potential when I say that I have a few single moment political 5 which means that key square in his village of momentum squared plus the 2 sides will Australia so I write that the rights that th the original momentum square close the centrifugal energy which is the energy linked to the angular momentum minus the potential intervention OK that's when all its solved showing questions points but this means that the effective potential is the son of the 1 square overall square which is positive and goes to press infinity and the minus 1 over here and the sum of the 2 if something like that I have I you always have to forgo bereaved I have I'm very large the at section of the crude Newton Falls and therefore our sacred albeit is when you sit at the minimum OK but from the diagram you see that if I use the Newtonian approximation here I will always have a secure albeit even if the wages even if the 2 objects are as close as you want they can go on simple but now if I stopped adding the stones here you see that have minus-1 overhaul but the idea of press 1 of the 12 square which of the opposite sex so it gives a visitor term which and then you have my motives when you do that you find that the becomes both audio and reliable in the sense that if I used to pin all the I get completely different answers at some ideas OK when the registers of the order of 10 gm I find that depending on the approximation I use I get nonsense or whatever but now we can remember also something that there is a limit to where we know exactly what happened it by using the parameter on new is the most social vicinity frustration when vicinity must small which is not the case but let's consider formerly the case where it is more of a small particle around a big the big masses of fresh and therefore I know what is the effective potentially effective potential is different potential for particle moving in Shoshone black and this effective potential as not this shape it is quite simple the effective potential flush shield is the following the it's all energy square and it is call when you solve Judaism interest you find 1 minus to GM I have put back in the gym here Jim wasn't what wanted plus angular momentum square instead of fighting some of the candidates seek the Center terms as square overall square minus Jim or I have appalled and because of the product and because this thing vanishes entitled to the potential the effective potential as a qualitatively different shape it is OK but do it because the advantages of 5 4 2 and so it means that this potential so it's normalized 1 at infinity so he goes to 1 it is like that it vanishes when the
registers to GM and then the 2nd all of its is this 1 OK but I see also that I have could all bids which is the maximum of the potential with this unstable securely so we learn that and then so industry Maples more you I know the potential should be like that what does it mean I know that the system is losing a bit of momentum so is the end momentum diminishes this thing becomes but I got an idea this is this maximum and minimum they marriage have an inflection point which means that's when the test article is a wonderful should back on their use of wages below which there are no sacred albeit subtle welcome the stable a with this I just is when you computing to 6 GM is 3 times the trashy alleges so this is saying I expect that the the exact to pollen thing might be all this type OK at the time when we welcome that there was no agreement on what you would expect but you would expect that quite effectively the thing should be over this fact and there which is another reason of saying White force Yukon cannot use because of the low so like the name of of the thinking the front the seats and the gift total nonsense that in general so this is the only way I have the inflection point is called the unevenness someone but I call it the last table albeit at no all of this was understood that in the in the 1990 and that many people try to go beyond this time there was a problem with posted tone and was not good enough for 1 of the at times it was done by a bad idea of such a percussion and and myself in 1998 and that they were 2 ideas initially we said OK let's assume idea but cities we would find it's not a good approximation to India but let's assume that the problem is to know about transformed these expressions In better expressions that incorporates the physics you know like for instance there is at last able Roberts soul mates incorporated in some way and to do that with the idea and set up a catch With ADI this soul but I can use that so this is the ideas by the way I already stated that nearly people most of the references to this paper as you will find in know as a as complete as links actually on the webpage of as there is a on iTunes Web page there is a gravitational wave page and actually cable of things so when you see names in blue click on it and then you have the paper what is the access to the paper and you so we're firing such a Cacheris said part of the prime is that this function UV which is directly linked to this is not good and that we should introduce a better functioning and we said the better function should be the square also modified energy and this modified energy we said for some reason should be this was a guest what put the fall over 2 1 and 2 the fall so he have the mention of instant square and all that so it's a dimension less energy sorry no variation of witnesses that we take away this season I think the now this is an it's an energy source it said mentioned an OK and we said Let's see place this extend opposed expansion of this function instead of the original function and because we know that this function as of now born in the show limits let's take a bed the approximation of this function for the foot for the following reasons there when I out of all these expansion the powers of the of some viable when I have the function as full Fulbright and the only thing I know is that the extension ideology professor what can I do to import there the question is a 495 opposed to expansion which is a Polaroid's pension alkaline pulled the function why out the coefficients increasing so fast why is the thing so bad that the merger the reason is when you think about it is the exact functions seen in the complex plane as singularities OK and that no if I consider function that over rivalry but and in the context play and I know that this function as a single IT somewhere in there Singh writing can be a bolder bunch good things like that but we know from complex function theory that the 1st single right in the complex then determines the edges of convergence and therefore we find no weather forcing writings I know what is the status of convention and I know what the other question cause because it does just of convergence is somewhere that saved single writing this means that the coefficient they are this means the series is like 1 of the into to the power and it's of this and then this means that for efficiency and almost like 1 of the 1st things right into the power and OK and any of these things is closer than 1 then it means crucial goal exponential OK and and this was saying we understand that there is indeed on some function is the Aliso single IT another function the right wing singularity because there is a segment singularity which is that the unstable albeit CA albeit they also exist only up to the end Beatles and there are no sacred orbits Apple and when you think about it he pulls that the function the flux functions for instance will have a single item at the light training in an image when you and therefore you expect it and therefore at the time we said OK so let us place and now I want you to eliminate this single right is to have a better function what you can do if you know there is a poll you can say that sparked all the poll but factoring the ball at the In France told us all it is to say that the place posted on Polaroid's pension by the ratio of knowledge from the federation is 1 blond plus and 1 of the greatest improved square divided by 1 person 1 vote planned 2 that is to say if you replace the power expansion by national reflection you can hear injecting formation all you can guess where is the boss and once you do it that's their decision to apply the approximate this fractals the ball and therefore the function is better represented that was the 1st idea to use the expansion football for the energy and the flux working and that the and actually eat work to some extent OK but then in 19 so that was in 1998 book in 1999 and the another idea came out to the dynamics better and actually idea became the weekend why I was the French version of the bulk of its extensive and I have checked that the it's not an English version of that is in the French preprint version there is a chapter on the Idol Jan Apell more energy levels and they say there is a work by preside 6 songs and Azinger stunned the amplified by further whereby which allows to his son nadal that ones for this thing and that at the end of this said good function energy function is this thing so when I saw that what was working for including 2 the effects or the other genital including this function that we hadn't introduced use for another reason the year before I said looks interesting and then we find some of what I know we pushed the idea further and then you'll be which was invented then so this is now what I'm all 299 the what is this an effective 1 by the middle of the Medford which will when the previous information is the following you have a two-body problem
which is this thing OK and other explained before this means I have to lessons and 1 and 2 that move around and they interact at the lowest although think the right by the airlines gravity but at although the right to know what's in the security would say there is a blob of Borden only 19 circuit Einstein equations and non-linear so the interaction between these 2 things contained a lot of Subianto OK and these you know how to compute by perturbation theory so it's 1 so this is equivalent to the both expanded and Estonian OK but the idea is to say we want to incorporate something which is where the formation of the exact case we know which is when you is 0 I know at this particle going along the slushy of Michael so in that limits I know the exact dynamics is a geodesic motion in the flashy metrics it's something nonlinear it's something number so the idea is to say let's imagine that there is some met union New affecting which she's not the Flushing metric but which is a matrix which depends on the monitor new and which is a continuous the formation of the French imitating the sense that when you is the old it is the fashion with for months CapitaLand but when you is Nunzio it's something else that you need to determine such that a particle renewal of you well you is the usual Newtonian effective mass produced must be having a geophysics dynamics which means the action is Table use the report in this effective matrix that you don't know and you want to say I want this thing and effective particle removing ineffective metric to be in some sense equivalent to the two-body dynamics that computed the pocket so this is the basic idea of this effective warm-blooded for formalism for the dynamics will come to that so it would be a way of listening in a sense the Amazonian OK at this point but we really need also to withstand the addition reactions separated OK where 1 needs to do so things well how to I said something but the world's OK also out compute so he I haven't known functions OK I'll do my computer's functions that the IGC is something I learned from John Wheeler which is this is a purely classical problems OK although as before we have seen qualifying and diagrams for physical things but it is useful to sing mechanical that is to say if you think of this quizzical component as a quantum pollen it simplifies OK why because when I have a two-body problem like that quantum mechanically it means I have energy levels and energy level you know etc. and press the D it's about quantum energy levels they are classified in quantum mechanics made to quantum numbers to integer quantum numbers 1 which is the the angular momentum quantum numbers and eligible is the anger of momentum this is as state which would be a good 0 his status and equal 1 and then there is the 2nd part of number which is that the number of which counts these things like this is the ground state and equal 1 and accord to is that I don't want them member and then you can define actually the the principal quantum number of the group problem land use and France an artist where analyzed quantum number is the number of nodes in the region quantum wave function you add N and you had 1 in order to avoid the zeal and think OK and as you know the old if you fall but I remind you that for instance in the corner problem that the energy levels are like 1 minus and Square became the only ban on at worth of alternation on the total principal quantum number OK but now this is for the theater this is for the 2 Poland which field but if I think of this problem the dynamics of a particle in some of the metric 1st you realize that because there is isn't so the symmetry look at this case he Ireland the to the fall of the square and is this is explicitly invariant and other publications include dimensions OK related and therefore this problem should be also sacred symmetry of can therefore I'm looking for a sacred symmetry metric therefore I am mistaken recently come into animals on the other side so I will have also levels I have a different problem and now once you say I think in terms of these levels immediately you say but the correspondence if I want this to be the same as they say I want this thing to correspond to this this thing to grasp onto this thing to this here quantum-mechanical of integer quantum numbers therefore it means that the effective and romantic should be called to the wheel angular momentum in the center of mass friends and their the door handle quantum number effective should be called to the they don't want a member of the but now we are interested in the classical problems so let me take the large quantum number of inmates it about going to 0 and going to infinity Watts is what determines this quantum numbers this is the boss ephemeral and the Balsam warned is saying that the and I pressed 1 of its vise is a call the action by both for the degree of freedom I have a separated the dynamics and D action viable in of PDQ on the cycle divided by 2 after all you don't divided by 2 you have each year with the age but you have to buy a and in the lounge of the integer number this is saying that these quantities which of the action viable II should be identified between the foreign so at the end you saying I want the real and the new Soviet missile as a function of the action by both of the real problem to be affiliated but related means what you realize also when you do a little computation that you cannot equate this energy to this energy you need to allow for non-military nap between the 2 and scale so let's be generous and say I have I have put away the unknown functions here so that's I put in new unknown function which is the energy map as so there is no known function as such that this should be called to the effective energy as a function of the action survival of the effective thing but I identified the arguments of the effective things to the affected so now he gives me integration of this described by the way in sinister mechanics the Etonian expressed as a function of the action variables closer by the pollen is called the bill Manitoba OK so sometimes recorded at the wrong and going and taking technically the reason why why please understand that we could proceed fast with decided is that With the heart Schafer in 1988 till 10 years before we have computed the DOE to pin the only and which isn't on the table calculation that made described very quickly so if you take 2 PM and onions you have these beasts look can now you want to say that we solve this problem exactly as the inflatable problem by uh by doing what I take that to be an idiom Estonian I go in the center of and within the center of mass and then I worried that the action please minus the energy the process as the old Of the afforded viable I survived a time because I have damaged the Jacoby equation which tells me that the real it is equal to GM you the realignment as a function of art and is the old idea that the usual no time independent and Jacoby Gration whites that the 0 OK this is a sacred Semitic problem so you
can solve the problem in a plane for OK let's I straight to because certify of fire you know you just fine and then you write that as the old is added and fight whereby definition and defies the angular momentum for angular momentum of the system process original action but the regular action we'll be of of this interval of where appeal is obtained by providing this in these which gives and explicit wish and so you plug you get kiosk where the skillful focused chaos seeks the because something you can't solve this equation passive activity as the functioning kiosk where by expanding in 1 of us is square and you can eliminate Ohio power so it is that is something which is that is the square root of something 20 you compute this something you find that it is the New Delhi approximation would be kiosk where you know is the energy pressed women were also you get a person to a more plus-size person momentum overall square but you get for atavistic origin this coefficient modify and then you have accepted on Monday 1 of the few facility to over fall close to the 5 of you get something like that you know you need to integrate these although cycle and this is aware that we've got so we discover that you can do this in the naive way of what's in physics we are told to them all you can go back to good old falls and in the Book of Somerset at them while they wept techniques for computing these things because this is what some of them at to compute OK you he was solving the problem before it is too correction he was the weights from the fine total consumption of was the 1st correction to the idol John levels in the Balsam of something and that at the time of the new out to compute these things and they said What is this interval this interval evidently in modern times we could say it's so it's considered society period but we don't need to say that it's a period of 4 and I barely takeover OK so the same thing is something I have a square root so it's a square what of a polynomial in 1 over so as amended by geometry here because of an but that's what you want is you I have this thing vanishes at the point of 1 . 2 with depends on the energy and angular momentum of the system for a given in agenda moment to end 1 of us is square there is this an easy all year and he and I want the interval between the 2 things I want this interval the reeling toward from this point of the spot this period but the take of some offense because it's very bad if you do that you realize you cannot do the computation before the board of the than 1 overseas you expand your over infinite and and I'm not part of the museum there it's very complicated why there a very nice trick you said this I can think of contour intervals the contour interval in the complex plane will actually give me what I want because I can deform the common to embark on this thing so it is the same but I have no doubt the ideas but there somewhere else so the contours close enough I can do that but once I am outside that I can now expand their waiting formally impose a 1 of us is where because I don't have single right using the prime is that this thing has single is clear the suicide expand the singer writers get worse if I may have no single so so we use formulas of this type and then we computed the answer and then let the and by the way is and this is needed for the 51 but if so what you get for instance explicitly relied dancer too the 1 clear thing we computed at 2 PM and then with the on skin Schaefer when we computed the CPN we computed that the run them infinitely beginning in 2000 2 thousand the classical wrote their own monotone is a function of 2 quantum numbers and in classical thing where n is what I define and I defined the classical analogs of and which is simply the this is called the president Bush is the sum of em and and as you recognize this is the sum of this in this case would you and what you find is this is an instance where when did miners 1 overview of square over a square so this is the quarrel the minus one-half of 1 over and square the famous cruel thing will fire the flat is due and new McCain I hope I did not forget some scaling is there's still no I think it's just G and new ideas put into it and uh care yes I think it is OK and then so this is the Newtonian answer very easy to get to know and the 1 you have a correction of Foursquare overseas square 1 overseas graphing and then you get to go efficiency over and then -minus one-fourth of 15 -minus new Nova and square OK and then this is 1 thing and then had to appear and you have more complicated things like the band which are only in 1 over and won over and things like that but this is an explicit expression and now how do you know what gets effective 1 body directions yes it's upon summation of the 2 Pinon Newtonian in their own name action now know let's consider this fall and so it is that the real live version of this now let's do it the run a version of this but for the signing to say What is this affecting metric can I compute something without knowing the function here you make and something which is as I said that we know that this is such a as to the strength of symmetric their false if it is straight smitten can write it as a general strike a resonant admitted as can change with age and go too far should coordinate their for this the espresso this square effective union you idiots and its new high white seats as minor a over the 2 square meters grand plus the over the elsewhere this where the come against plant where the the Michiganders field-effect square Poseidon the Pesquera defies square appear as usual motivation and I go to Foshan coordinates we have where therefore general isn't admitted his bomber twice by 2 function it this list of exterior styling independent I don't know it's that look at this the summit on is autonomous time evaluate you look at the dynamics the symmetries of India because it's that's what is confusing this thing is not that metric that each about cities where local because it's an effective thing at the dynamical level but no what we know as we said above this function or should it should be a function of new and should be reduced to the Shoshone so you know 2 things OK 1st introduced to Foshan some women and also as I'm doing everything I suppose you to expanded it should admit deposited expansion and therefore I can just say this thing will be the 1 because it was unknown proficient it 1 that I have to determine GM overseas grip on another provision into a new GM's overseas where all square press etc. If you continue this the over new should be also 1 person duty they want new GM over his square of personal use the tools etc. The Square opposite thinking the it's the free this Cuba at the time now I have reduced the problem to computing coefficient they want it to be wondered to up so it can expect a similarly for this function I've because this function as uh let me ,comma twice it's immediately I can write universe I can write this thing saying that the effective energy divided by Musa square which is a dimension this thing should be called to 1 person the appeals binding murdered a nuisance square times 1 Chris as someone the video binding program used
square personified to this square with etc. where I just defined the binding energy is just the analogy -minus MC Square treated the energy of a litany of usually you forget about the INS's quick here we are 1 the mistakes that we have to remember that Chris somewhere so if I put it after subtracting so he just returned the most general function which reduces the Newton approximation sings the same energy model of the new as an end in ties already missed a correction with unknown parameters are found 1 of 2 but now the game I want to determine the oldest parameters by some question what is the aggression and the question is this so I need now to compute the general non-Newtonian all of this effective dynamics with this so what what I do is I write the unbeaten Jacoby creations of 1 lights New Square the new effective the new unit Pinon Canyon where this is the way the action for the affected problem accords won't it's the match condition like embellish it's the simplest way to solve the geodesic dynamics by Milton Jacoby but but if I play abroad this year have the same type of equations that he had the same summer from tapping but except that now they depend on laboratory coefficients wanted to be wanted to but I can't do the calculations and then when you do the calculations you find that the defective that energy as a function effective that is to say as a function of all these unknown coefficients is a as a function of the 2 major numbers on 2 would numbers of the affected problem but I identify them to the and easy call to because it is a test particle of must knew the 1st term is not his news square contends that sets the flight which shows that I cannot identify this to this OK I need some transformation but progressed I have doubled and the Newtonian Paul online audiences the answer would be the same because that's a if I put a priori that say this is the quantum the Honourable so the defective problem that put plans but that will identify the pines to the unemployed the 1st then please just the it's the Newtonian thing is that there is a 1 one-over all potential in both problems he comes from this term because as we know this is the new approximations that gives the 1 over often and then yes no but then I get correction 1 plus square overseas square and here I get instead of numbers I get combinations of the unknown parameters like for instance the 1st where I and prime time I get 18 squared minus the tools minus 1 half of it is the 1 today 1 of which is considered description and the 2nd Commission is minus the 2 they want minus 7 . 8 it is 1 over and plans quick and accurate in writing that I think that's I have already assumed to be a description of the 1 here that's that this coefficient is his to which is just I at the new approximation of these it is just given by this stuff so I need to identify this provision by the usual intolerable to nations and then you find that it is the 1 most people to mind to it's a convention is just aware of and it's not totally convention because I've put the sum of the 2 lessons of the it's that the Newtonian interaction is proportional to new times the sum of the 2 masters the public often 1 and 2 which is equal to the anyway it is 8 1 from the new to approximations Busby miners who sold the unknowns this 1 at at at the fines of the wanting approximation this is unknown this is unknown and this is unknown so I have the unknowns alright the other 1 pinup automation and now many questions that I love I can identify this and these modulo this equations so it give me any aggression which introduces a fear of the unknown which is a 1 so actually I do want to an approximation of the unknowns which I have found 1 1882 and the 1 and I get to equations and this 2 equations there this so I would not repeat this sequel to this no I wouldn't be because its modified from oneself 1 iteration is it 1 square as it too of minus 1 of 450 1 the 1 6 so this is the effective upon this is the real problem plan I have this coefficient on the way to the 1 minus-7 over a creatively the 1 but when I use the transformation have an extra correction coming from someone and when you do the competition you get behind us 1 of the one-for-one and this is must be equal to this coefficient over 4 miners new colorful where can they won minus 2 OK so because of this I have 2 equations for the unknown at this stage although we relaxed at this thing and we looked for more general solution we have a long discussion with let's make the further assumption that not only the question of G 0 0 in this effect image reconstruction like but also the coefficient of the of the regional thing that is to say let's assume that the rise gravity Of the effective retreat is equal to the Shoshone gravity in 5 days justification for this 1 camp over it is a necessity primary but let's assume so if I assume this now it says I am assuming that beauty is the 1 is the of the coefficient of flushing of the term 1 over which is pressed to now with this 2 assumption is that I have seen it because it is not a for the affected metric units USA that's a lot so under these 2 assumption now so I know we 1 so you see this is this question for 42 to now that can domain and now I can the determine of someone came and the results it is you find and that's where things get so simple that it too is can't afford 1 is neural to and now let me comments and that ideas and that stopped that's that's the end the the what does it mean stink this thing means that I had to do so at the wanting an approximation this term is the old their full and an end and the don't sound Shoshone so this is saying that the 1 an approximation I have a test particle removing in Shoshone but won't and that this particle removing in Shoshone but won't undoubtedly on the condition that the the transformation contains the fusion which is used to so energy map these equivalent Sosa geodesic interest is equivalent to the 1 PM I'm staying trim so this unemployment which when you have to work with it is always more complicated than it looks OK which are full different coefficients is equivalent to Foshan so it's the 1st simplification of the UPE things became something which does not look at all like Shoshone because this thing is equivalent in disguise to at this particle renewal moving in dishwashers thing to 1 kid who can't and then after lunch we will discuss the younger 1 kid OK let's stop you In past the masses you always the mass of the particle to all of would be in this case the is the sum of the 2 massive list yeah it's and chemical and the sun is very simple it's like in the Newton in case is a start but it's at 1 PM leveled right as
Einfügungsdämpfung
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Metadaten

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Titel 3/4 Gravitational Waves and Binary Systems
Serientitel Thibault Damour - Gravitational Waves and Binary Systems
Teil 3
Anzahl der Teile 4
Autor Damour, Thibault
Lizenz CC-Namensnennung 3.0 Unported:
Sie dürfen das Werk bzw. den Inhalt zu jedem legalen Zweck nutzen, verändern und in unveränderter oder veränderter Form vervielfältigen, verbreiten und öffentlich zugänglich machen, sofern Sie den Namen des Autors/Rechteinhabers in der von ihm festgelegten Weise nennen.
DOI 10.5446/20545
Herausgeber Institut des Hautes Études Scientifiques (IHÉS)
Erscheinungsjahr 2016
Sprache Englisch

Inhaltliche Metadaten

Fachgebiet Mathematik
Abstract 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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