2/4 Gravitational Waves and Binary Systems

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2/4 Gravitational Waves and Binary Systems
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This crash course will review the theory of the generation of gravitational waves, as well as the theory of the motion and radiation of the premier expected source for gravitational wave interferometric detectors: binary systems.
Point (geometry) Axiom of choice Standard error Functional (mathematics) INTEGRAL Transformation (genetics) Multiplication sign Function (mathematics) Dimensional analysis Derivation (linguistics) Group representation Wind wave Sign (mathematics) Positional notation Lecture/Conference Different (Kate Ryan album) Term (mathematics) Well-formed formula Helmholtz decomposition Scalar potential Square number Modulform Maxwell's equations Damping Nichtlineares Gleichungssystem Representation theory Set theory Condition number Area Multiplication Moment (mathematics) Algebraic structure Infinity Thermal expansion Price index Vector potential Vector space Symmetry (physics) Order (biology) Gravitation Summierbarkeit Object (grammar) Coefficient Identical particles Nilpotente Gruppe Thermal conductivity Directed graph Spacetime
Principal ideal Functional (mathematics) Transformation (genetics) Multiplication sign Translation (relic) Mereology Grothendieck topology Product (business) Derivation (linguistics) Group representation Mathematics Sign (mathematics) Population density Lecture/Conference Term (mathematics) Well-formed formula Different (Kate Ryan album) Scalar field Operator (mathematics) Conservation law Nichtlineares Gleichungssystem Position operator Representation theory Condition number Rhombus Theory of relativity Block (periodic table) Moment (mathematics) Thermal expansion Price index Sequence Numerical analysis Arithmetic mean Symmetry (physics) Vector space Gravitation Summierbarkeit Right angle Object (grammar) Identical particles Limit of a function Directed graph Maß <Mathematik>
Group action Multiplication sign Equaliser (mathematics) 1 (number) Parameter (computer programming) Mereology Order of magnitude Grothendieck topology Derivation (linguistics) Mathematics Analogy Oval Thermal fluctuations Square number Conservation law Körper <Algebra> Extension (kinesiology) Theory of relativity Block (periodic table) Moment (mathematics) Price index Sequence Infinite set Vector space Symmetry (physics) Summierbarkeit Identical particles Sinc function Resultant Directed graph Spacetime Point (geometry) Functional (mathematics) Transformation (genetics) Distance Rule of inference Spektrum <Mathematik> Degrees of freedom (physics and chemistry) Population density Lecture/Conference Well-formed formula Term (mathematics) Operator (mathematics) Modulform Symmetric matrix Set theory Addition Multiplication Scaling (geometry) Expression Counting Line (geometry) Mortality rate Loop (music) Logic Factory (trading post) Gravitation Object (grammar)
Building Group action Differential (mechanical device) Multiplication sign Direction (geometry) Wave equation Parameter (computer programming) Open set Mereology Order of magnitude Derivation (linguistics) Group representation Sign (mathematics) Mathematics Plane (geometry) Positional notation Velocity Analogy Thermal fluctuations Square number Conservation law Series (mathematics) Körper <Algebra> Position operator Potenz <Mathematik> Block (periodic table) Closed set Moment (mathematics) Gradient Price index Lattice (order) Symmetry (physics) Gravitational wave Vector space Angle Order (biology) Right angle Moving average Identical particles Flux Directed graph Spacetime Point (geometry) Functional (mathematics) Momentum Divisor Transformation (genetics) Polarization (waves) Frequency Wind wave Population density Propagator Causality Lecture/Conference Well-formed formula Term (mathematics) Average Helmholtz decomposition Operator (mathematics) Energy level Modulform Nichtlineares Gleichungssystem Condition number Module (mathematics) Addition Multiplication Wavelet Consistency Projective plane Algebraic structure Limit (category theory) Sphere Numerical analysis Vector potential Formal power series Film editing Vector field Doubling the cube Plane wave Combinatory logic Gravitation Object (grammar) Musical ensemble Coefficient Local ring Maß <Mathematik>
Group action Differential (mechanical device) Multiplication sign Direction (geometry) Kontraktion <Mathematik> Insertion loss Parameter (computer programming) Mereology Food energy Order of magnitude Grothendieck topology Derivation (linguistics) Group representation Mathematics Plane (geometry) Positional notation Many-sorted logic Scalar field Square number Conservation law Körper <Algebra> Extension (kinesiology) Physical system Process (computing) Moment (mathematics) Computability Infinity Thermal expansion Lattice (order) Price index Variable (mathematics) Flow separation Sequence Gravitational wave Symmetry (physics) Order (biology) Summierbarkeit Right angle Identical particles Flux Resultant Directed graph Functional (mathematics) Angular momentum Momentum Tournament (medieval) Mass Distance Polarization (waves) Power (physics) Product (business) Wind wave Degrees of freedom (physics and chemistry) Population density Lecture/Conference Term (mathematics) Well-formed formula Helmholtz decomposition Operator (mathematics) Energy level Modulform Nichtlineares Gleichungssystem Hydraulic jump Module (mathematics) Multiplication Standard deviation Surface Projective plane Total S.A. Sphere Approximation Numerical analysis Combinatory logic Calculation Gravitation Waveform Object (grammar) Coefficient Table (information) Pressure Maß <Mathematik>
Logical constant Group action INTEGRAL Multiplication sign Direction (geometry) 1 (number) Sheaf (mathematics) Parameter (computer programming) Mereology Food energy Dimensional analysis Grothendieck topology Group representation Estimator Coefficient of determination Spherical harmonics Positional notation Many-sorted logic Velocity Square number Position operator Physical system Area Potenz <Mathematik> Theory of relativity Moment (mathematics) Computability Infinity Thermal expansion Nominal number Connected space Binary tree Gravitational wave Symmetry (physics) Vector space Newton's law of universal gravitation Identical particles Flux Resultant Astrophysics Spacetime Directed graph Polynomial Functional (mathematics) Einheitskugel Variety (linguistics) Characteristic polynomial Mass Distance Rule of inference Theory Element (mathematics) Power (physics) Wind wave Goodness of fit Root Causality Lecture/Conference Well-formed formula Term (mathematics) Complex number Energy level Harmonic analysis Modulform Set theory Noise (electronics) Multiplication Focus (optics) Military base Forcing (mathematics) Projective plane Content (media) Basis <Mathematik> Incidence algebra Sphere Approximation Numerical analysis Quilt Universe (mathematics) Gravitation LAN party Waveform Object (grammar) Coefficient Maß <Mathematik>
INTEGRAL Multiplication sign 1 (number) Inverse element Mereology Derivation (linguistics) Sign (mathematics) Mathematics Different (Kate Ryan album) Square number Körper <Algebra> Extension (kinesiology) Theory of relativity Moment (mathematics) Coordinate system Infinity Thermal expansion Perturbation theory Summierbarkeit Metric system Directed graph Spacetime Vacuum Einstein-Feldgleichungen Transformation (genetics) Nichtnewtonsche Flüssigkeit Theory Goodness of fit Wind wave Lecture/Conference Natural number Cylinder (geometry) Term (mathematics) Well-formed formula Energy level Nichtlineares Gleichungssystem Set theory Condition number Time zone Addition Multiplication Forcing (mathematics) Content (media) Sphere Approximation Minkowski-Geometrie Calculation Cube Gravitation
Multiplication sign Kontraktion <Mathematik> Water vapor Mereology Grothendieck topology Group representation Derivation (linguistics) Uniformer Raum Different (Kate Ryan album) Square number Körper <Algebra> Stability theory Computability Thermal expansion Lattice (order) Measurement Angle Vector space Order (biology) Right angle Figurate number Directed graph Arc (geometry) Functional (mathematics) Einstein-Feldgleichungen Divisor Product (business) Power (physics) Wind wave Lecture/Conference Complex number Helmholtz decomposition Well-formed formula Green's function Modulform Energy level Nichtlineares Gleichungssystem Analytic continuation Multiplication Distribution (mathematics) Dirac delta function Convolution Formal power series Network topology Universe (mathematics) Object (grammar) Table (information) Weltlinie Maß <Mathematik>
Group action Matter wave INTEGRAL Euler angles Length Multiplication sign 1 (number) Kontraktion <Mathematik> Mereology Food energy Dimensional analysis Subset Explosion Derivation (linguistics) Sign (mathematics) Plane (geometry) Roundness (object) Different (Kate Ryan album) Velocity Square number Extension (kinesiology) Physical system Social class Area Logarithm Theory of relativity Moment (mathematics) Computability Infinity Thermal expansion Binary tree Arithmetic mean Gravitational wave Order (biology) Right angle Summierbarkeit Identical particles Spacetime Directed graph Point (geometry) Functional (mathematics) Einstein-Feldgleichungen Finitismus Link (knot theory) Divisor Nichtnewtonsche Flüssigkeit Image resolution Tube (container) Mass Distance Event horizon Graph coloring Theory Power (physics) Product (business) Frequency Wind wave Goodness of fit Lecture/Conference Term (mathematics) Well-formed formula Natural number Divergence Green's function Gastropod shell Energy level Modulform Nichtlineares Gleichungssystem Analytic continuation Condition number Time zone Addition Multiplication Scaling (geometry) Validity (statistics) Forcing (mathematics) Basis <Mathematik> Limit (category theory) Approximation Numerical analysis Vector potential Residual (numerical analysis) Formal power series Calculation Gravitation Waveform Object (grammar) Game theory Coefficient Asymptotic analysis Table (information) Local ring Maß <Mathematik>
after the head of a and in the end of the world in and it issue of the Internet and we don't want to think about it and would so I wanted to to finish indeed gravity apartheid was a bit fast so I went straight to the multiple expansion of the 20 cases which is technically a simpler it has the same problems as the currency while there is a gage transformation there is a moment supports and what you want an idea and you have 2 sets of multiple moments so let me explain this from scratch so the multiple expansion of the Maxwell equation resources b so we are like gravity we impose a condition the Romans conditions for pulling it isn't "quotation mark 0 then when equation becomes this summer and fall piracy Jue OK the only difference so far lunch signs this aid becomes H-bomb you knew which was there before all of my kind assistance is the things and the souls of doing this is to new and that as the let's remember also something which is true in both cases there's a compatibility condition evidently if you impose this condition and with this thing is because the source was concerned because the finality of the divergent of the left-hand side has to be there but we know it was yourself said before I did anything because of the structure of Maxwell equations that on the structure of the new if new new accord now final the sign for from this thing and there is a young kid entity which says that the fighting ingenue these resources at 2 p 0 which is well it's also integration accepted the contracted Duncan lunchtime contracted until identity for the unch tankers now whether or what I said is that if you decompose the source Jr New that's a core of in the charge density with the conventions in the veracity of here and continue although there is a special current similarly knew contravariant issued decomposing in scalar potential in the vector potentials OK then as I said this thing you can solve by using this general formula so did that the skater potential of fire is sum With this decorating coefficients Over all and forms used to infinity of special derivatives so I remind you that this is the repeated special derivative at times referred to in the and then here you have a cell of you over on where European cleaners so what we are doing here with me and is that we have a source in which is due this source is generating waves can you take a point of view .period here and you decompose here in in this form in terms of symmetry Traff create services in the symmetric jurisprudence serve which is recovered wave also because it's a function of off with this derivative it's a solution of the of this equation outside the salt and as a I used 1 oversee the -minus above a 1st for the last 2 years of here you can use the formula ball but you said index like and that X 1 to 3 and but the multiple the composition of the composition which is symmetry expressly respect to these areas Anderson's can and at this stage the point the formula that I do not remember this morning that I have it here now is to say this objects these it is irreducible it is a signatory jurisprudence resource in the reasonable representation of the alteration clear haven't object which has citric we will respect of these Anderson's error but not with respect to this exciting but I can Composites in 3 but you Idaho 1st you put just some numerical coefficient I will comment about American conditions in moment delivered shootout answer you can't touch wilderness objects commitments wandered over to a press 1 at that time I do and minus 1 that made the Victoria Mark I wanted to make uh 1 advantage of working with symmetry tryst with is that everything is it is optional when you work with 4 one-year lambs order coefficients of square what's all final press 1 and things like that was a normalization conditions make square which appear everywhere and here it is algebra a month and sells with on press 1 induces and you just start of fact also come from the dimension of space all to impress 1 comes from some of the dimensional from India's spaces not what are these here I have written the explicit formula which says that the subjects belonged to the offensive of conduct of the space of vector halls and the space of irreducible tonsils with and indexes and this has been the composed this season and the symmetry jurisprudence or if it was 1 of the society is different from the 1st one here there is an objective which is a signatory jurisprudence with answer which I was this notation I did not defined and minus 1 yeah I said Ellen isn't the index means I 1 and to minus 1 is a convinced notation just to say I had 1 comprise index and minus 1 it means that it means either minus-1 investors but that the multi index characters so this thing as minus 1 industry is here and next indexing the notation when in the service like outputs in these markets this means it's support Jackson to make its symmetry choice we sold his
1st to make it symmetry then you take away all traces remind you that if I have for instance the tension with just 2 of this subject means 1st I make it seem attracts wanting symmetry part and then take away 1 of the on climbed during the 20's consists of it's an example of projecting tensile which is not symmetric and we entrust some trace onto something which is a signatory without trace and and this so this subject as ailing this is a symmetry trade with it belongs to the and and this subject belongs to the implementation diamonds 1 because it is symmetric ruefully and when muscles and the this is an identity if the objects which are out of the noted the other following these objects symmetry trysts with ease as I said that I take the object with EPA's 1 innocence and then I projected this year to interest principal ejection operator the city which has not only fairly in this service is obtained by taking my object G we decompose they saw this as Empress 1 in this and so I can write it as as G In the present then and minus 1 in this is that remain and then I've taken epsilon that here I have I want to try and minus-1 so I can add the as you and I mean innocent explained banks maybe and then I take this unit which will be projection over and uses for this subject is hinted to witty and is explained this is our contractor so far and the last object is the simplest the last objectives he said which and minus 1 to disclose is simply obtained by taking this thing and taking the choice but that I cannot take a price over this and this is because it is restraints on the only twice I take is between 1 index at the beginning and ending next them because its symmetric anyway so it means that I am minus 1 and this is remaining of focused solely so this is a very trivial thing this is a terrible thing this is more subtle and here you have an identity where it still appears twice you must appear twice because everything is fighting variant in what doing so if I change the sign of tsunami this convention changes signed the other operations fine and I use the identity of the product of 2 it's along is some of their stuff and so this is just a nudge a bike identity which is that the translation in the language of utensils of this famous saying that the representation of of control of Spain won ton skins spin and press 1 spin and spin and minus 1 OK just to vote no so I can prove these here but now I need to use tools different facts there I am in the gage I can add 2 India I can make a gage transformation he I mean magnetism soldiers consummation is in danger of a scalar widening gravity it is the kind gentle for vector with tools in literature and and to stay in the Lorenz gage defining the role the means the new is your would then it means I must have a block of time a 0 so the most general engaged transformation I can make to stay in the same engaged and to be compatible with this expansion is to set aside a general solution will be the sum of the health of the South Island off the miners pharmacy overall so I can play so I haven't infinite sequence of the tryst with answer which is major to political position of the gage transformation so if I had that this term here I will have a formula which the and on the various so now I only look at the irreducible objects of symmetry jurisprudence so this was by this formula it's is given and a number of this fellow there he is yes let me say that these objects are obtained by the solution of these equations as I said this equation is of the same type as the scale of the question except there is an exciting index which Presnell which is a numerical index so I can use this formula used to compute it found the these GAM as a function of the source of coastal like for instance FM would be any with be this interval into all of this create off the source the fall 5 the source for fighting is the is the density of 0 OK taken in that sense of the bar and then accept so so it looks as Fairland this stage looks like the multiple moments of those charged density but this is not the end because I have this all the things which are are you an CNN and the end for nearly all of the day and across was 1 listens and cabin for generic and they are signatory to assuage any concerns which are also given by intervals and story and there is the time of the year which you know I put to you the Abdel-Rahman's this thing with this interval of the but no I have these objects which are intervals over the salsa and then I have this idea which I can do whatever I want to simplify the formula and I have something else which is that of the Salsman be and the conservation of the soul uh which implies this thing if I applied this formula here I will get relations between this because you see if I take the time data 5 and display the narrative of this I will have multiple expansion of the sites you fight plays the sigh with government support expansion in irreducible terms and you feel something inadmissible terms something was vanishing each irreducible thing as to avenge so sold this condition of this condition gives me the relations between the time derivative of offense because in doing that I have so I am also an identity which I can't even the rights of the people said yes that the write it so it will be clearer for instance I get an identity which is the time derivative of DFL is a call to something like over its excelled in England is at the very top that and then of my the divided excise Jerry plus Z and I do time that evidence of G T yes and then this thing I can
integrate by parts and then from these I get
relation between some of these moments here the ones that appear here and then 1 needs to to work out all these identities and use outside so that at the end you'll find success during the the explosion in the summer yes he other you means to change papers and amendment to it's a viable and uh I'm trying to get the logic of the thing so that's after gage transformation which you computes so you compute what you want to add OK you compute the site by using these identities between the offense and the other lamentable movements so that at the end of a new gage unified pints of TMX which is the son of former the 1920 the factorial and the Q L of the miners controversy over all but went to recently and he I find in this day and age see Annex is minus-1 oversee the interval over felt asserted the summary minus 12 reflecting also this is repeated special Bharati refund the and minus 1 in the service of what happens is the time derivative of this symmetry Tracy on these objects of symmetric jurisprudence idea and minus 1 C minus of on our side since players 1 over the other 2 were responding to interest rate can sorry I'm reading the long lines where the things to be here it is less than over this 1 a I in the deep it and minus 1 I'm so sorry to have to find a formula :colon it is this petition calls to members of the city divided by 2 so let us look at the result after luggage transformation and using the conservation rules relating to the fact that I am in the sources concerned so what you in this distance formation there the general rule of of Atlantic field is expressed in 2 sets of multiple moments the Q L 2 full charge leads a charge magical moments it is the analog of the magical moments of the charge distribution but artistic context and enemies for magnetic it is magnetic multiple moments and the explicit expression of these objects all that Qs he is what is the good form a line here to have a better formula here they the a key to them 1 of the qualities I have exactly the counting is that they wear fall set of multiple moments and a because you can count on in Phoenix a sequence of multiple moments like a block of 1 of these infinite sequences killed by additions formation and the other 1 is viewed by the consequences of this provision which is Cayla discrimination as knowing this is so engaged uh when I expanded than in Phoenix set of relations between the derivatives of all this fall things and indeed Thank you Natalie the counting is that I have for -minus 1 for additional summation minus 1 from the conservation and and and the same thing in gravity I will give a full time small gift transformation and full-time small Conservation Law identities and then this eliminates then multiple moments minus minus-4 for begins at the end 2 sets of multiple moments that's why uh editorial season like gravity at the end as 2 degrees of freedom because another way of saying this is this number of of the big sequences of magical moments counts like 1 scared of him because remember once Carol Field is an infinite sequence of magical moments so 1 sequence of a stiff political moment is like 1 scandal 51 wanted his skillful we know that clarity as to the degrees of freedom like electromagnetism that's why at the end of each degree of freedom scale is like an infinite set of multiple moments and that's why you expect by this and counting that there would be only to set of multiple myeloma Florida languages it isn't explicit expression is that there is an interval of Varitek's there is going to go over there between minus 1 and press 1 and then there is this is the 3rd of the experiment and wall in the science where so sorry yes there is what I mean is this thing so let me put it today where Paul today means that the time of the month is shifted from the to presented the exodus you don't shift the space argument you should time so you would reside inside and it's it's an operation once you do that you will Jose that appear then you need to integrate ovals and this is a function it is the 1st time so this term looks like charge density which some fluctuation in the time argument multiple moments like a possible moment of the Johnson said but that's not the forensic because you have this is what you wear opening for FM actually the formula for I fail is what I will fairly is this but it's not yet the physical magical moments because I needed to do these initials mission conservation also there is the next from the transformation and in that case you find that the extent of his minus born overseas square To help was 1 over 1st 1 impress think that there is no doubt that which has now an index and plus 1 as a function of that and here you have accepted a L uh did object to being sold at a time of agreement still would be denied is the time derivative of the Covenant which is a vector also here you have effect of which is contracted this index is contracted so it is a stiff because of this sense of fact why he I was scared and it is stiff because I have this thing so this is like Connecticut take tied to its phonetic take pride multiple moment it's difficult to build abandonments and you have a magnetic stripe magnetic diet which is a mail all of its arguments OK depends on 1 it is very similar for me employer for the case of communities in the trial of Z and here you have X a minus 1 M 2 days of my fellow hearts which is also where an is the magnitude station than magnetism Asian density means it is a vector of other vector also it's a bridge thanks Paul G. you have you have so we are talking about a point isn't so there is a chance density and there is a current density you take like the magnetic mode of equality and like all bales of looping current OK uses a small loop which
is like a small magnets or magnetic moment the infinitesimal magnetic moment of the current OK and the miniaturization density which is effective field inside the and this vector field you takeover multiple moments and just make it the symmetry transferring so this formalize actually very simple the only certainty is that at the time argument series shifted by something so all activities attic stage OK the the publication of the velocity of right of the thing is soft follicle magnetism at the end module gives transformation that's why there is a prime the 4 Atlantic put their full potential outside the source is expressed her the eyes derivative of this moment all the time derivative of the same moment because of the identity and display the retrieval of this magnitude density and that and and this formula as if you look at Jackson Irish Open the Book of Jackson you don't find this because everything is done if you assume that things the band like the exponential I only get to that they are monarch function of times and then use the and functions and things like that so we can look like a bit complicated and bits a function while actually everything is in Dubai and the only subtleties to have this function he whole but he then there Is there a transformation which means vessel function has come from the co-sponsors of the becomes a reality explained it comes from within I attorney United cooks mysterious that's what it looks like eternity it also looks like a violation of causality in the sense that this is saying that administers this season this formula saying that if I have the source here and if I compute the field in you at this point in space this thing does not depend only on the source at the town of time because of the because I am fluctuating at by not by T minus on but by the press although still minus so I have fluctuations which looks a cold and the reason is the following is that the multiple political position is not the local things in which a political position says that if I am here I think this Fiore had the same ideas on and then on this decompose the police feud between the union and you civil thing we respect alterations on this and therefore what I'm doing is I idea I combined a few original order although all those cuts of the source entering the definition of limited political position as much a political position nonrecourse here and therefore each uses an average of the salsa which as this double Lycos torture II need to think that the solvents in Tunisia became so I need all of those averages in this thing is that can and what happens is that for high-value surveillance this becomes sharper and sharper than your closer and closer to the like anyway this is the this is the answer to connect and gravity I was Whiting soulful gravity I will not be widened the formula my assistant they raised but if it does it but the important point is that it that again only to type of multiple moments we must fight instead of electric type quoted must and spinning type with evidently the analog of magnetism is that is being targeted and these moments I would just be Cordell notations there where could call I am and Jeanne actually the letters J is chosen to evoke Spain with his other spin moment and then you look for letter close to NI is better than case solidified his Senate testimony before so this is the idea that this means that there is a combination of blows which is with 1 polarization of all women in the nation and also noticeable to polarization this this thing uh that they should we writes in the incoming I will willingly the team gravity so I would like to think of it in ,comma His this is why it that I will not find a loudly on the question of just 2 of the structure I am not in a gage the point I will build to explain is that I am not in a village where I see the 2 pariahs editions of the gravitational waves I'm engaged and I have more call I have I can still do additions formation to project on what like go and get will consider and this is what is the next thing I want to explain enamored engage aged in the 0 0 rolled up I would like the only this 1 because it is the son of a failed this year just for fun and and this thing I T minus the over so that's the simple thing plus the gage their no HBO it contained something where I don't like the coefficients there is the minus 1 the time Nativity with this guy with this I and minus while at the same argument menace of divided by all and then there is a 3rd term which is at the low and high into the ayes and minus 1 James B. and when 1 of the many of the overall so at the level of the I component of the metric I have the time derivative of this must notable moments and I have space narrative of the magnetic spinning type multiple moments and at the level of the treaty in space if you have something weird and minus 2 Nativity of actually you peel off it's a it's a very simple thing here that there are adding this is Bedell a contractor and this gives a skater he I Walter vector also want to phase in what you do is you peel off 1 index side but on the symmetry with the thing and as you want to know the derivative of a function of Timoner-Salva Yukon tied the other remedies you take away to innocence but you need to attend any meetings to compensate and the interviews and on and you use a conservation earlier in doing that you absolutely need to be in a condition where this is satisfied because you use this to relates to the magical moments so which with 10 multiple moments minus Foreman this fall which left to multiple moments so here you have to attend their it's this and then there is in the Cottonwood coefficients and then you have the dead die a sorry I missed normal sources of this formula then you have the as you expect period J and then you have J got that is to say we have epsilon Amy i j . index j the and minus to the function of the mind of the the Over all and in any of the verities so all of these objects like me repeat on the objects that appeared to in the elder the type a certain number of space that have achieved and then some functions which some the of T minus part sold all the building blocks of the Dublin waves there were some special limited their solution of the of the
creation freedom creation and the objects that come in out of irreducible representations of the global politicians and you have 2 types of the same IRL appeals time differentiated twice time differentiated and this so so this is must die movements political moment and this has just been typed 20 Beaumont OK and the former life as that is what I had the hidden before which was the here because I think that it was here inconsistency I rely just the 1st time like I had of its arguments couldn't you is an integral of space or it's going to go into all of our desire that the size of Texas and signal at the shifted argument with all the terms where signal that was this is what I would like to be followed with 2 0 0 0 plus overseas square OK so it is a must .period moment and the other 1 under the momentum density of flux in the Salt have in thinking and he just came here in so here we are a nominee gage and indeed I have chosen the gage but now to answer the question of care if I want to exercise this expresses given the source it expresses the fear that it admits but now I'm interested and we have some interferometry here and I want to know In this sense I explain the fall far from the source it should be a plane wave OK 1st to answer a question I forgot about the form of a goal the when I have so let's let's look again at these 2 of these clearly when let me do even an example if you when I have 1 special derivative of something which is careful to minus to be and what is this this is minus its belief that the space they retain acts on this idea to nose and eyes overall square at 12 Clemens plant OK but the special limited can act on the hour which is here and there for having accepted which is minors as adults at the argument of the minus sign off the gradient of art itself which she's and I welcome the unit vector divided by so and if I have molded over teams I more time there with his over but I will always have determined 1 overall so when I go very far actually I mean only this time with a light and then sold the if I go in the in the waves along very far I will have something which looks like a plane wave which decreases like 1 of her on the coefficient the 1 over our here will be obtained for instance by taking at the David uniting 1 this guy everybody acts on these here and and here we have everybody should act on this and then I will have this plane where I have part of the thing which is a place where 1 of our know what he's obstacles by your gravitational wave detector as we said before and to decompose in a different gage this thing should be equivalent to GTT because we have this morning that given a general solution of the Eisenstein equation very there exists a transformation of gage where have special components wells it was 0 is I want to go in engage with agency was enrolled in the world where knowledge by use in the fall of the death of a thing where the victim parties 0 and only the special about IG is this 1 0 and then but still I mean Lorenz gage and in obligation which means I have determined that Senate condition that this thing would be 0 OK and and as we have proven this morning and the traces of which would be the also staying within the ominous gage I can know we find the gage divided the island gage is absolutely not unique because I can't change Monday engage a lot and here I want to change it in order to extract from this formalize what will be observed by labeled with this is what you want to compute the the the points there a wide explaining this is that this formula has paralysis to compute the multiple moments that are emitted but now the next question so this is what you computer-mediated good is what do you observe when you have like Google and now I will answer this question I need to go to these gage how do you do that the way you do that Is that 1st and that's clear here if I go very far from the source I we love the uh 1 over all wavelet but local leaders 1 overall waves is like a plane wave as we know locally very far from the salsa into like a plane wave and what did we for plane with people that the plane wave in Genoa looks locally or something like that there is a phrase factor and then there is opportunity and that is that it engage in that it engaged you pulled checked everything in the middle of son if you have the source now the going is not in spacetime but in space so here expects have the the sources this is an exciting way to prepare this born of the decomposed OK now I look in some direction him and I say locally these waves here will be playing these are spherical waves but to say we're very far from the source troops like a plane and that what we have pulled these even and what is the direction of obligation if I'm at the point exuded the original publication is an entry is the unit of from the source to where I am this is the wave of I'm here so insurrection and their therefore what I need to do is to pull object on the plane what we know is that many the waivers obtained is only chance to the direction of propagation so there is a tenser which is symmetry to a suite in the roll playing on this he tangent to the sphere where I am all too little or too and so I think I as a unit vector I think that to play in all the world to rent and then I need to project a this wave on display in in a form which is the symmetry cancer without trace and and this is the answer in the sense that given the direction and I can define apologetic tional Prairie PIG organ which is that the Jake miners and I'm injured OK this this is a politician operator in the sense that if you start on the vector falsified correct all the PIG managing the politicians of the is what it is that say it is the policy actions her can be projected in a long and can and then perpendicular to it and what evidence is the part where have subtitled What is perpendicular so this thing Paul drinks Perfecto of arbitrary direction even effect October too during the direction of the wave now H I J T T is obtained by doing this projection twice because you pull this that is to say you go from this village to it engaged at the end of the day you find that if you want to compute what like gold detects here OK what is what is the wave the wave comes from some directions OK I can't take it as something existing in the plane perpendicular to the direction of the wind and the libel detector or is also at an angle so I will have also caused signs of objection on leg over this is a separate calculation and compute the to polarization of the gravitational but I have a client server so what they should do is that I think age many for instance in these gage I take it it's 1 of us so I think the 1 over on a amplitude of the wave of it's a kick in the manager of the I think these pieces the plane wave upon this formula and now I want to put checked it as a dancer in this presents what you do inspected this projection of 2004 the direction and the G fall and this project depends on the player but it estimates trace so you need to supply the trace and therefore the projection operator is this minus 1 of of I J of N P K and you just hold this OK it's only the final whistle this is an explicit as undermine homeland which says given the you know we
actually need gage occur where but we're satisfied and shiny equation in which his weird like I think it's special components and then I do decide under to buy projections where n is the direction from the source to wear but this formula has allowed 1 to compute explicitly uh the OK so now I can combine or formula that is to say I take this formula which expressed the wave a wand assaults in multiple moments and then I ask What do I see and here what and then you get did you gets really did the composition of what you see in the light of as a function of the source because of what I have sold by this formula as is that I can compute the waiver mutants 2 orders in think I made Nova slow-motion approximation located the source can be more tolerant and think it has to be a millionaire I've got it so the next difficulties with this to go beyond the life of the city and we will see to if he did the right thing to do if he said on the 1st day of the age this has got to find a long time the the "quotation mark nobody I have I have the quadruple jump to Portland magnitude and yet those although we will get this this is nothing new ideas to wonderful effect so let me 1st defines a the file that might interest in arise gravity let me call this a here this is this is a projection operator or book which contains for distance because it's so let me call this thing a P before soapy care effect it said it's another by cooperation which takes a cancer and projects eating something simple which hasn't components in the plane of to the direction and that they chose the formula which is so this is the 3rd time just raised less than to polarization of the gravitational where there is a full of fact for which comes from Justin Romanization days OK so what you do you think this formula as an I said You keep only that the part of the way which is in 1 of OK so for instance and only H I J because finally all the other components of the Belmont important for the project and they must be related in some way that's why I I treat the same object that appears here so immediately as here we love letters to their eventually this guy and that is when the original but as I said if I think the 1 of all Part I. We love and minus 2 molded Havertys because I need to act on arguments minus also acted as if I am at the multiple I will have added that wanted OK because I have to derivatives plus and minus 2 so to present some names that demented board and will enter with and the evidence of the callable relented wish to derivatives the tuple with the dividend is at in the way for the so but that at the end you find that you have this projection of Britain's pungent primed Jr In the sorry and changing the notation which Spencer admitted the petitioned the unit but just denoted capital and several of small men but it's the same as before is to distinguish the feud .period from the source .period current head you have a sum over know when you do understand you find that there are multiple moments for an equal 0 0 and 1 that disappear Y because for instance the multiple moments for liquid 0 0 is the total mass of the system which appears like a Schwab she's an overall OK uh but it is not my adjective OK so so this term when you do that the Tutsi expansion this wonders appeals a similarly for the the Didier angular momentum of the source which is a constant quantity in the center of mass thinks so the physical or other degrees of freedom that you recall that Infiniti this starts with the quadrupled the amount they take all men must being that then you have an infant son where also of for the moment I have kept the velocity of light located looks like a new reasons to to keep the veracity of life but actually what is important and this is in a sense what Einstein already undermined is that the multiple expansion is at the same time and expansion in powers of 1 overseas if you take higher multiples the contributes less and less in the sense that they have become the was the see that's what the probable role with meaning it's the meeting all term and then the opted porous a small correction 1 overseas Square formerly compared to the probable for etc so you get this thing they hear you of objects that have been on display in mine too so cleanness ,comma no is just capital letters and minus 2 so -minus To help oversee and this 1 and press 1 factorial excellently B the Jr the administration and they and minus 2 so contract here OK have so In the the mass multiple moments I would define what is the subject of a at this stage I am defining the following things I'm saying that the the 2 polarization of the gravitational waves can be returned as an infinity it's the sequence of bore the moments of the city so these objects are symmetric and parametrized lies the magical moments of a general where focus so that to type of multiple movement but the work we have done before has expressed this things in terms of ideas and which themselves fights price in terms of the source so actually I would be able or like I'm standard and the lowest level to express what you excel at infinity directly as an interval of the source the Nile there is no approximation has been made of the small remove you receive that me the folder must die performance because these are must type moments the power 1 overseas and why there is a next couple were 1 overseer press 1 for the magnetic thing because always the magnetic spin effects in diversity as the next time receive compared to the thing so there is less admission there is always demand for instance the magnetic type quadrupled is less important than the must die quadrupled that's why Einstein Formula Ford Mustang audible was writes it is the 1st of if I write this explicitly just to see what's going to days and so it goes like 1 of our own OK so I chose the notation several times but that what is smaller arguably comes capitalize just the distance away from the source
and here I am taking only the coefficient of 1 of our by using this formula has and I watched the film relates physically and before I become too because there isn't 1 of them here so I this projection so 5 plunge which depends on the direction from the source and now let me write explicitly the the beginning of this thing yes sorry the resorts often arise gravity is that these objects you are so symmetry has this is finally equal right to the end time variability of the mass multiple moments where this notation means as a functional bond-by-bond the and find evidence so it's the end of the 10 to the power of ideas so it's just a notation when I have a function of time have because you can you can whites 1 . 2001 it to adults when you have to there it is but when you have adults the convenience of you need to know how many times did so in the arise ITT this you here is just me and had anybody and I explained why the reason is Kate appeals with two-time derivatives have and miners to spare the liberty but the act on this thinking on the plane with the and therefore it is finally ended the narrative of the muscle the same thing for all the year as a function of the Illini arise reality this season the magnitude of the being tied multiple moments differentiated again and times you can see that it's the same number of differentiation Ukraine together with the former lands which gives that Ireland is still means a lot of the source for no I have an explicit formula is being identity which says if I ever with some of the new news In the source continue I computes the annotation to the 1781 he will press the assessed square concluded signal because it comes in that you wanted by the way it is the combination which handles all of our cosmology cosmology you of all procedures because this thinking is the density press the times the pressure the care and the fact that in cosmology in some form realizes this is important is again important here for inflation for instance variable because it changes the site is just saying that Einstein's equations yes I'm actually going Simitis quantity and terraces is that if you look at of the 0 0 for much questioned the source here is your pressed assess is so hopelessly which appears on the right inside of which is not of the extent of sewage that's why it comes in here so let me I I was writing explicitly to see what it going so this tournament using this formula and summing over it gives II during which 2 derivatives so now I the quotable moment which to Nativity This is the Einstein terms to the inability of the political movement except that they didn't Einstein had said that the possible moment is given just the 1st on here at the usual quadruple moment why here we have already be the quadruple moment which many already to corrections so even if you keep on it is that you have a mall but the next is 1 of the city of the fallibility of poverty watchable moments contacted by the Victorian from the source which answers here look at the process full of efficiency that you know and I will stop after system change In the end the OK wished to deliver the soul the in the waveform the poor moment offer being typed and at 2 crucial 1 overseas more of than this that's why I'm I had said that the West approximation I wondered about the board but here we see quite a portal to Poland the 1st spinning type of quadruple and then you have infinitely I the don't like that to square for instance the next term which is the demand for index a magical moment is something like that could you just more and more of an effect also more and more time derivatives and things like that so this is the result of the the arise gravity and then from this resort Yukon of already compute for instance what is the energy flux and uh the angular momentum flux when you do that this is where they are and at this stage it's not clear why it is useful to have a decomposition in the ensuing retrace freely objects and the ways is that when you computes the energy flux for instance the computation of the energy flux is essentially will be going to pretend to be it's not because of the soaring energy flux I mean all of us feel that is to say what is the total energy flux invalidation away emitted from the source essentially this formula is you have table or overloaded sphere all you have on the surface of the 2 sphere of the H I J K T . H I J T he ducked the square ,comma module coefficient of of the wave which is the 2 polarization of the waves are and they are like the energy useful portion or to the time the liberty will why because as we know for scalar field for instance the communion office Garifuna is the new fighting Pfizer Inc and cost minus 1 of I think that it's always find activity of the field that come in the energy OK so similarly although I don't want you to enter into the discussion of how you define the energy flux in these waves by the way in the book which was sealed this morning of Madame showcase this is also discussing the high-frequency thing you define the team you knew of the year Award of the gravitational waves of the lighter side of Einstein's equations when when you I can shiny creation and you say on the right hand side I put money now terms coming from the right show itself the contribute like an additional term to tuneup and this gives this at the time a lunchtime Einstein computed even before getting what is the price and serve adult and sort of guarantee yet conservation also use that in the end the undergoing formula that you could not compute general momentum but you could come to the energy from anywhere so imagine now doing this calculation what do you do you you you take this subject you take a time derivative so immediately each derivative here Wildmon by 1 unit the political moment will have 3 times the narrative this 1 with a 4 1 thing and now you take this square and now the fact that these not symmetry these are simple thing means that the cost products or vanish because when you have irreducible representations the a October of refers back to their natural and to that you up and therefore when you compute the square you will have only the square of this the square this the square this Musgrove this that I had the knowledge OK this is the nice thing so that for instance for the energy loss you will get a lawyer for the group
efficient here way out but I would like to the sense of the thing like the energy flux said fluxes in energy he is a son as with some numerical coefficient of if I take these objects this would be you we've wandered into the square plus he added we won the ability of this thing square which some numerical coefficient and if I place this formula UAL is the Elf that ability of all of them and 5 the most magical moments this means that summer L starting with 2 where I am of the mass political moments with ailing this is differentiated so it's good to try to help us 1 Times Square which some numerical coefficient plus the Jake L. differential Empress One Times Square with an American coefficients and the older lunchtime quotable formula consists yes and the now if I keep the woman with the square I have the island 1 of a Cecil knowing it will oversee square and actually if I put everything I have this 1 of us OK so it means so that the news the they get there this call like 1 of received too well and this cause like 1 the receipt will restore OK this would you overall reflect our so this is saying for individual well done magnetic thing this being thing is less important grow overseas square but it also means that the ultra board is 1 of us a square smaller than the probable and the the couple exactly Coppola as well as his query even smaller and and and this is very important because this is saying that when I have a binary whenever source before characteristic velocity the overseas if the oversees a small parameters the multiple expansion is also directly an expansion in the overseas I know that if I want to compute things up to the overseas 6 or something like that in the energy flux I only need to computer up to a the for moment or whatever and I can't neglect everything of high multiple moments and and this is the way posted a method can compute things that nearly to the end that by including more and more things although there is a lot better with the last thing that we you come back to the and so now have talked for 1 hour or so later stopped for a break of 15 minutes and I will continue with the nonlinear thinks know I we sketch of old ones because everything I said was to give wall was to give some tourists because the stores will be useful for the nonlinear film because imminently we are not interested in eyes gravity denies guarantee is never a good approximation in astrophysics because as soon as soon as there is a system with internal got additional forces and we have talking about a binary system so it means the nonlinear effects of gravity of the ones that created the force between the 2 objects and therefore for binary systems I have never the noise about it so even at the lowest approximation that's why actually the paper once dined with the probable formula cannot be applied to a binary system it can be applied to an oscillating border 4 reads something like that but not to a system which is held together by gravitational forces you absolutely need to include nonlinear effects of 180 even for the lowest although political formula for binary system 1st and this was realized 1st realized by non-dollar initiated in their treaties in 1944 they explicitly say we need to include terms and at the end when Ewing phenomenal terms you recover the same thing looking callable formula lunchtime and and I want to go beyond that because this is what we need to do To Ohio approximations OK let me just give that's somewhere and I have I will add 1 formula here which is just to make the connection with what he's done in this facility In light gold type things is that when you have for these things OK what In practical terms a what what people recalled the multiple or the expansion of the gravitational waves he is is measured by some quantities which are called which cast a pall ejection along the tensile spherical harmonics OK so there is a theory of say ,comma monarchs which generalize the scalar spherical laminates and I must say it always looks complicated but the the actually the result is very simple to understand because when you decompose the tenser GTT along stretch tensile fake or harmonics which I labeled by 2 integers LNM like the usual rules of spherical nominates with and growing for minus 2 per cent OK I I of just saying what is the relation because what what you find in textbooks and the quantities I have introduces the relation is very simple Is there is Newton's concerned is the distance to the so in the waves or there is a square would to for some historical reasons like the film constant and things like that it's the only square which which appears there is a woman received a helpless to OK then there's a 16 by over OK this is not important to you just to know this and then you have to numerical coefficient that because now we are in the realignment of the normalized spherical nominees was interval of others feel tense aristocrat is 1 you will have square Woods for the 1st time because each time you have a wider lens you square root of something so here you start thinking through what is just a convention before we do not have screwed To and minus-1 1 but what do you have here and you have the UN which is this year and cave which itself does not contain screw it promised to do I find that so you Ellie is must type white Yellin you see the notation is always to have the 1st letter is the more important must die and then you go in the alphabet and it'd be later like I J users to have something that you can understand and then there is the 1 of receipt and the level of the amplitude basis quilt of inputs to over to and and was 1 and miners want you out of here and then here so he and these estimates expressly 1074 adding this is why they say is a complex number it's a project so now you project the symmetry trace will object on a busy so you compute the number like components something because yes maybe that means any goes for minor sell to personnel as we all know OK this means and thanks to impress 1 values OK but to impress 1 is precisely the dimension of the space we talk about care and therefore any objects which Instinet twist
reform is a symmetry to this weekend so we fared in this is is a member of the irreducible representation of engine to oppress 1 of pencils of Acadian and therefore there exist many bases on which the contents as useful information only to 1 components will carry the components for you so so you can indeed there is the same in formation in innuendo and something we just element this is OK but now for the waveform the convention is too the computers in the this in terms of this by adding he this where I hadn't use this this morning remember that the usual why and so you take the textbook with wire and use a one-year lending as a function very well the unit's sphere we Feingold that the fight I can ,comma 28 in terms of unit vector and we've and scraped 1 and actually has explained this morning it is a polynomial in and end this thing is wire land God and simply felt it is a polynomial off and for there in the components of an element the compliments and McMeniman you annex and why I said with NZ costs the dog and he had this assigned the task coastline and assigned the task assigned 5 so if I the place this in I will have a high we lover exponential I am fine OK the usual thing and I will have polynomials inside the tank was sent it out of the little type that there would explain this morning anyway this defines this thing and I've explained this morning this subject for given LAN users Semitic jurisprudence for foreign investors and the 2nd is to win press 1 and so is the basis of the sites of symmetry themselves you used as bases to transform the symmetry trace returned served in a number which is the components in this and embassies and this number with these coefficients gives you a challenge and this is a challenge is that is what is often put them in the cuticle computation of for instance the main quotable wave is etched into this incident even in the most wanted to be the case by new ideas that he mentioned that he I was saying uh when the source is moving slowly the multiple expansion is an expansion in powers of the overseas because what is surprising is the science is that when you do it is a composition for crediting black or even near the end of essentially H 2 2 is the most important thing and the other things are smaller and they become compiled boat which 2 2 only really very near the coalescence and their contribution to the energy experiments will so in that sense this authority the fixing of the reconnaissance of 2 records as the good taste of being not to on idealistic nearly 2 so so this type of Formula reviews but now I want to explain something that is a section of the for the station it's the is so that the good thing is that I my assistant did not erase is everything I can answer now because everything is on the black hole so dictated that so let me let me indeed I think that mean all of them the summarize thing so I don't need this that made me put this up and so something very simple happens unionized everything which does not happen at all of the money identity but we read the news that part of the thinks so In the linear arise guarantees of suggest In the arise gravity they wear the types of multiple moments in the sense that we said at the H-bomb you knew the gravitational waves of this was a formula written somewhere here is expressed In terms of falls source multiple moments so it is a function of golf to set of multiple moments where I just sketch is an integral role of other sorts of paying off the mass density X and the focus on the Croatian and things like that so im this was something I compute from the source like if I a binary system I think quadruple moment of the muscle the neutral solve most of the mutants are focused on the salt and this expressed the wave in Majie political position but now I had also sold the wave at Infinity I transformed to a 2 to gage I have the wave that goal is detected this wave was expressed in terms of how did magical moments so http which is computed on this story it's not used these Xpress into notable moments but even if I am in the nonlinear theory I can define all Ellenville not because the formulas would not be all quite so I can't say that if I in the notoriety I take the wave at infinity because this form of that is just the privatization of the where radical was intimidated and therefore it does some additive magical moments now in arise variety we have found for the UL is simply the time that we've achieved an offending material and the area is simply the end of administrative of GM's tho this is a violation which is valid only in your eyes about and it will be the modified OK yes so I had those those are the things I can integrate the somersaults I can't Palmer the solution outside the source in terms of these multiple moments and then I have idea to multiple moments which opined that material is so I know everything you know what happens in the in the pneumonia case he's much more intriguing but 1 can use steel thing so I was sketched the technique we use so this is what we call the MPA multiple Opel's cause concern now so multiple outposts with Coast Candide winning Yukos governments not pose and no you need we are going to do an expansion which will use and and this is the idea of this expansion was 1st proposed by this 1 OK in the restricted setting and keep farmland eyes may be its usefulness and then we have taken the place of the thing and pushed it in there as a tool for computing things in the morning way so what do you do you are going universe also which for instance is made of 2 now I know it's a space-time picture of the source I'm really interested in is the motion of 2 black also support to neutron stars in space so they make like this the book and the move this and where you want to go very far from this 1 billion light years away he and you want to know about in this direction what's what is the H T
TIG wave that's you get here OK you the way you do it is you can still use this formula as in the sense that if you imagine that they think of the sphere 1 billion light years away on the sole hostages would be difficult but you decomposing set in this century trysts with answer and you just Parliament adviser way you by this by this summer could just a parameterization but now you want to know what is the relations so you have this UN Indian desire moments at at infinity that you can observe them and to answer the question of Nathalie they're all gauging their this thing I have fixed things so that it's getting there and these are like the 2 was of freedom of the wave but indomitable now you want to know how can I computer knowledge base in terms of the source knowing that I am not tingling arise Garrity I need to take money now affects many nonunion facts in gravity so how do you do things you do things in successive approximations 1st you got going to decompose then you decompose the spacetime in various zones declared around the and then you are going to use different approximation methods in in each zone there is a zone and the big approximation method other than the 1 with the big demand over the with me this would support was all the notion of a Moreyra approximation method which will be In the big cylinder which was essentially to infinity and the reason why does not go free to infinity is that there are some real guys their and you want to modify them because you work in a nominee did yes this would be done in a nominee gage now the Armani gage is bad that when you go in additives on very fast because the other terms that pop up and that is the date victims down not very difficult to handle and there is a the paper of 1987 by Russia to to show all too dear to introduce Bondi sacks died coordinate system so that you I have a good expansion in Powell's so you need here you will have some coordinates that fight in this region you would need to change the coordinator of unknown according to absorb some of the guys make things OK and to have a good expansion of the ball the sex but also it is part of the scheme but that there but most of the calculations there would be 1 here and here in the zone near the source you will use force Newtonian theory but later OK because this is a convenient way to compute the fields generated by the other sources the the the the mainland the meet thing in the hand is what happens in this thing and there were so the scheme is the following what you do in this exteriors on services and the exterior of the 1st you outside the source so use or Einstein equations in a vacuum so you solve on you knew you could 0 which flat but I'll do you do it you do it by a postman Corsican expansion which means in general it's good to to work now with the what was called in the old days the which means you know square with the G Jimmy New inverse content and you and the Gothic matchmaker has it and you knew that many this there is if I want to keep this thing if I was pointing here this way I think if I put the minus inch about the stage by unionized theory is the same as the before because the the perturbation away from the goes metric for the Gothic metric is actually the rich Munion minus one-half inch in a church in the calculations you don't put a minus sign here you put it somewhere else OK that is not worry about just assigned and that use which equaled 0 0 in terms of the the advantage is that when you are a nominee engaged these subjects so let me now forget about the signs and the the would presage a new page menu is not limited cultivation is this or if you want you can put like a good each now that there are many conditions said that is put to the the conclude that this is your exactly so the gage condition that is exactly that the divergent so this thing is your because of evidence of it new is equipped with you know when you write Einstein equations you will so when you write on the new recruit 0 you will have the questions that you can write formally answer that number of this it's new is a sum of terms which contains to David and then I have to have quality can manage to derivatives have terms can be contained each and then to the anonymity and then quantity nature McCain I I I 1 just white and shiny equations formerly like expanded on flat space that we know it's not going to be or was very good but each time there will be a problem we relocated and then take into account the physics of of what happens that because this is not on various charges a flat space-time not only in this way you can solve wheelie explicitly that good postman Kostka because now why is it called multiple Opolis Mingo skill because at each level so you are going to solve each news as a successive thing saying his G H 1 union Mustang on all their injury which to new cross sailed all the cube new new principle you you push the calculation of 2 the where where what's H 1 you will be the general solution of that on their offer which 1 you knew it was you but this means that when you knew is likely to arise gravity-wave outside the assaults we want to get the field which is using monetary conditions of new incoming ideation so we look for a general solution about integration outside the cells which is pipe with but we have solved this problem it is it is given by this OK there is the general formula for general ingenue except that as well known in there we need to write that each 1 new news these so let's call this thing this formula that can only be called a solution of Einstein's your eyes gravity expressed in terms of 2 set of multiple moments of extended lamentable moments that we are going to put you out not related to the source because we are we it the source we don't know how we are we are somewhere here we are just bomb advising the thing so the meant aboard the 2 sets of magical moments which can only call New Korea we called them to distinguish them mn Anderson but just to remember about enemies must died and has since been tight OK less like this and you need to take into account additions formation because the general solution is using the general gage and before changing gage it can be very mean you need actually to take into account later some changes of gage so you need to put back the various uh I forgot only that there are some gage 24 moments the ones I said we we can inmate's OK sorry there is nobody you therefore that's what I said before there for coordinator
transformation you can make so there are formidable gage transformations which are their you
put them but they play a minor role but it's important for the skin to Canada and now now you you eat every punch by accretions explicitly and and the power of this method is because you will be able to solve of Einstein equations explicitly because you combine the postman coast expansion was because can just mean these so we have done nothing at this stage but you inject an explicit multiple expansion for the solution and now you look at the solutions offered me right here at the solution for this thing on all the bits you have this thing and in the right hand side when do you have you have to universities and you have something quite erratic in H 1 but H 1 is the son of function Of the measures over the summer "quotation mark pitch 1 is of the that you take this you are the more they've achieved OK if it is a time that evidence put stability if it is a favorite in the field months the number of and so the sense torture but it is not their souls to compute explicitly this known enough things 1 way is to expire they we think because as we explained before this thing is like 1 of our effort with the and delivered the plus 1 of our square and some unit vector if we if and when 1 David the and things like that he sees at the end what you get is something for the explicit and then you need to to do and it's not that you argue it would you decide you compute thinks that this thing is finally which contain by worlds of 1 over on the general review of the 1 of our time there would be a function of 2 miner's son because you have products of function of minus side and then they will be In this is OK so depending on this you will have new new witnesses also will floating around and then OK after doing this thing and I are here yes this that means this function of Minnesota is the pond so here it is the product for instance of a man with some derivatives the miners are crimes difference said mn too the T minus I would 1st have come about everything which some contraction of innocence so you know these where you had his civil representation of the additional but now you make the products of 2 inadmissible representation that like the and 1 kind of 2 OK the new and then you need to do legal pose it even in a usable parts OK to have something simple actually what is most important is that the Angler parts you can always when you have angered all parts you can decompose it's in and hats and because there is just a formula for intensified and I injury I can say this is an agenda which is a signatory trees plus 1 of them Taiji so OK I just use this to orders so at the same level and therefore you need to solve things like that but now the difficulty Is the following this is a function of the miners are so it is a kind of wave which was like that and it contains the universe follows of 1 but it contains very high interest powers of 1 overall because if you think just of the public portions of the probable contain 1 of occupancy I for instance of the minus side if you take the time they didn't even you take the where you will have 1 of last 6 things like that and then entitlement to boards the goal like 1 of our 6 and what you want you want to to solve the situation now if you try to solve this equation in the form of the Dublin by computers the Green function which is the Green function acting on the source but what is the source of this also is something which has an angular dependence but it doesn't major independents OK and eventually opponents is divergent when you will formally through I yield which is OK because I outside the so I should not go inside I could see why should solve this problem keeping outside the source but that the power of the method that is that you don't do that you formerly known forget about source you say Let's because the idea the following day have replaced the source which is physically there by by a fictitious the world line which has dented functions in writing and reading material that of functions in writings in the sense that this the thing is a meeting multiples of multiples thing I know if I take that on behalf of the end of of the minor on not on their off as remind us of previous minus-4 by half of 2 of them have free access to its inventor function and it's a distribution fee reform along the famine disillusioned the therefore when I take and anybody I really have any special narrative of the delta function so it means the much aboard composition means I have place the physical source which was figure out which existed by a formal sites objectivity was collect of their material that the function something very singular when I come to the politics of the singular things that increases to Wilson go to infinity I get and table which is infinite OK so all these intervals -dash meaningless but this is where you start doing some technological thing which was inhibited by what we and doing before for the 2 PM problem is you use continuation he said ideas other mothers not rookie year so it is Schwartz like all died and continuation USA let me introduce the power of our because there's this thing diverge because ideology yet it is a careful with this this Green function what is this going function the Green function that 1 is using which is this thing this thing is is a convolution operator With the is a great function between the field .period T X and the source .period that safety .period ex-prime OK and this thing is what deadly than before it is that without the man with the prime minus X minus 6 . 2 the divided by minus exports so and you integrate all ex-prime so you have a have person like well over expires expand and humor to buy this by something which goes like 1 over at Pine to something free he diverges because the article or at opined "quotation mark 0 becomes insecurity the ex-prime of opined the press something and this thing does not exist a divergent so what you do is bigger but shall not interested in this story are interested in solving this equation which is a different problem so you have the right to do whatever you want so that at the end you get a solution and and what you do is you could see a factor which is onto our the water to the power of the Web is a complex number that you are going to choose if there a part of B. is large enough this gives the sense that the table now becomes convergence because you see if I have anything all that give an example 5 the the acts over opined Cuba because uh I go to borrow called in it's this means I have it's put out to the power and and let me change explained to this thing is asked where the onto honor guard the angles and divided by to the OK so I haven't table over the and then I have been saying all of the arc all the and minus 2 OK if any is equal to the all-out General This is the guy making divergent or worse but if I multiplied is by all of us he will probably this becomes a convergent in they will not could you'll is that we
are part of the is large enough
and therefore I can't compute this thing as a function of the Parliament of the now you pull that this interval is available fix so it exists in the so in the plane of the viable and which is a complex liable you go sufficiently found the right so that you can compute the interwar here and then now you ask is this the is this an analytic function of being the answer is yes because it's legal to look at it and that the only single will be poured so actually buy it's a continuation it with final mail Norfolk functional but at the end of the year day what you want is to go back to be equal 0 because this way you will have a solution of the problem you want and if you stayed away from the Brazilian whatever solution of the problem and that but the worst that can happen when so you you you continue to be up to the quality of all is that the worst that can happen is you can have always had a good view but it's clear I mean when you look at it you find immediately that the the was accorded the residue of the Poland the coefficient of I'm in the singular power will be actually at the the 1st although you get only a simple reports 1 over and uh down not always fall summit but if there a bullet would be 1 of the and the question of 1 of the immediately you is an emerging new solution of the problem OK so you can subtract it and you if you subtract the single parts you have what is called the finite fossil now you add the it's a finite part but not in the sense of a member of the finite parts because know where this is going to put a finite part for the 0 0 which when you take the long expansion so this this thing is is a function of the which as you know coefficients men being to case prices coefficients being the minus 1 the 1st the 1 over a B and then there's the 0 OK and then the departs affordability of the finite parties this you just subtract everything which is a appalled olmert to and and then went to pull is that easy always a solution of the problem you wanted do so is just a mathematical trick and the reason why it is used for is that you can compute board isn't it was explicitly and and therefore this way given any I think we all although often on United you have intercourse which you can compute your products of a stiff concerns your function of because that was the reason why it think simplify a lot is that all of those things are just function of T minus all and then there the power of our and then there'll angular dependence it's not like a complicated function that you don't know it is something of a polynomial essentially modulo some function of 2 and and we risk we could develop an explicit formulas for computing this thing at the moment in our level OK so which means that even in the order you can if you want to compute things you need somebody very energetic likely to pursue this due to high rollers and at the end and thanks to this work and energy 1 got was also at the moment in the level of care and now but the logical point is the following itself I start from this idea of unknown seed multiple moments so at the level of which 1 you put some unknown multiple moments OK then you computes the dissolution and the next all that which is a function only of these multiple moments like that but now with none out of and then you can continue so sold this way look at but at this stage you have now to where we should be nominee gage so we have to check is the only gage conditions satisfied after the finite but it sees modulo something where you can go right so you can generate with some of tweeting something which is at each level of solution of the young engage condition and the solution of Einstein's equations therefore attitude so you do that so sold this where you generates an NH at any multiple although because formerly you can continue this to then the gets very very complicated so you push things on to to some level but which is a function of unknown multiple moments and the next step is to set up but I'll do I determine that this multiple moments because now I have computed the exterior so 1 computes the exterior trickier outside the salsa as a formal functional the also when you do that you never to oppose you to an expansion because opposed Newtonian expansion is valued only Medanales on why it is crucial here to have something which is valued in the waves also you want the validity of this expansion to goal from outside the source of 2 Infinity's sentencing when you wrote in guy in terms that you do with and that the next step is to say how do I know compute the value of these things crucially these 2 moments in terms of the source rivals that of the U.S. has had this was always have had chairman of the moment the arbitrary Department finds a solution and I know this lot the owners of the houses as for the religious right here on this and it's not on the list then it was all that mattered extensions of it's you compute things when you mean the behavior that is ahead of the holiday the the issue is not what what you have to pull it is you have to prove that to all all because what what happens is that when you put for instance if you keep the number of Monte 6 said I go up from any quarter zeal to win court and next to make things clear when you do that the although divergent said the ideologies will increase at each holder of the post Mickelson approximation because there and and to be able to pull off that there was a nonentity continued on and then will gave me a real guarantees and then pop up and you have to pull 180 continuation in presence of rehearsal we introduce spaces that we call the lens which of the special signification for me but anyway and that was just the name of a which contain low them up to Powell's the end of the Road either and you pull that out and then goes to an empress 1 it and then you can do would report to oral those in expansion injury that you can't always do the continuation and you can always solve the problem formative so he generates solution at the end of the Einstein which reality is that it generates the general solution in the sense that this solution complaints that the number of arbitrary multiple moments that exists in that you expect to exist in a physical solutions that at this stage a constructive solutions and the next step is to be related to the source and the way you do it is Yusof now a different problem is that you saw the Post Newtonian problems you know you will reintroduce the source so you start with the I Have Nothing Compares which moves along and now I am going to solve Einstein equations by a different integration schemes which is opposed Newtonian scheme so opposed scheme instead of losing the battle bear thing you right down there has time minus-1 overseas where the group spread and you treat this as a small term that you put on the right hand side and Yusof Pressel type the polymer of the pose of the post on in the scheme is is very good for computing explicitly things to some older but it gets bad when you go in the waves that is same as the main event EDT formal demand of entity which is limited to be at a distance from the source are much smaller than the wavelength emitted by the system OK but the exterior of computation is valid in the zone which formerly ease at the which is larger than not something link to the women because you made no sales on approximation Heliopolis Maine coast and so the only thing is that you should be much larger than the size of the soul that's but is OK and if no the size of the salsa he is the much smaller than the wavelength emitted by the source but precisely this is posted to importing nations saying the velocities are smaller than unity and much more than unity you have an intermediate zone between 2 shells 2 tubes you know you have to have an attitude and in this June the to the 2 formal so these our formal of policy former expansions here the outside expansion is an expansion in powers of G and the inside expansion obduracy Square is an expansion in powers of 1 overseas square and then use
the method of much asymptotic expansion which is to say in an intermediate sold the 2 expansion should be expansion of the same object the cemetery a Maduro additions formation because we don't know if there are many gage we use outside is the settlement engage with use inside that's why you need to put the general gage because otherwise there would be a mismatch even within the wheel of a common gages and that this information you can at times explicit computation to compute this thing as as a function of the source close to the post Newtonian approximation computes everything in terms of the soul here you go .period everything in terms of these objects which are known by saying that the 2 things are the same you compute these things in terms of the source let me too and it was said this at the 1st TV or the thing which is that the 1 PM approximation to the question this request the political analyst with the fate of the 5th of and it's only now theory so it's a little of over the Majeed years the IRA isn't the answer was exactly so I did not need to it is because I have all the 1990's of Einstein equations so intensive died on this is saying the sourcing and it's something which is then the combining with something as from the source and then I have QB purchasers so I need to take the interaction of too much applause from the source that generate something outside and I have many even when his vision things to take into account their 1st I should say before because the 1st thing we did we break was due the first one pursue telling an approximation and there was something remarkably simple happening OK which does not happen too high although when 1 adds to compute the Tupi an approximation found a nice which is the following yes let me let me right maybe there at the final will find performance the so he to I don't see your hands OK the final formula as these you want to compute the motive force so let's call this the additive might support and this seems really want it if the way that Infiniti it's bond advisers what you see in the detector at infinity but the problem is when I not in the in the gravity I could relate these To the time there would you buy and I and was until over this also everything was now I will have more money in now terms I will have my I'm looking for relations with U.S. is related to summon the will of the source but which takes into account number now effect in a systematic way and they entered a method gives states and that we show the type of formulas you get because you need to distinguish 2 types of among nineties that coming but it may widen the type of formula political do you get that the the thing you are interested in each of the magical moments at infinity and so there is a U.S. and a view the new White 1st of the new head of the air is accord to so there are several notable moments in the game for the moment I have said there is these moments that can be called the I'm going to make multiple moments that you introducing the solution before you will lead them to the source then they disappear after a while these moments are replaced by something in terms of the source and let me like the formula at this stage you find that the UN that is given by a formula which looks like the 1 wonderful in the sense that this is the Einstein David table something that will be expressed in terms of the source but I will I still have to work but I have extra corrections which part From something which even knows very well which is the fact that when you I'm cursed space-time the Green function does not propagate along the right but it doesn't tell you that the Green function as a part inside the light gold and that these things when you compare them explicitly means that's what you observe at infinity Is that what you observe at infinity we're not the only function what the and delete all the time like going straight to the source to the velocity of land but will contain an integral role within the past that is to say what the source did not reveal kneels before we so it's so befall the light came already have been of after that that's a 10 billion years before stealing influences the gravitational wave at infinity so lacked the so-called energy eatery effects we call the Nunavut so you have intervals over the the past and this can be computed explicitly the 1st at lowest although you get this same the local press infinity and Donald of the same object so the among and means adding this is a and pressed to derivatives broker at the time you the the same user at time OK it's actually I should have kept the thing that it's not small you scare people knew the reason why it is covered with you is that the issue is not that easy is like the miners are minus 2 g the total mass uh pressing their love of our like you know in the Croatian gave me correction that there without the time you have to take into account the fact that they're like columns on not flat but they are balanced by Gagne terms by the total mass of the system and other things so you have this type of corrections that you normalized its way but this is still in the sense that the the real without a means to recover primary in the occurs this time but you depend on this something before we fertilizer maker wrong amendments not too long natural log can see Over 2 conceal plus 2 numerical coefficient "quotation mark 1 and the island of you is given by the same type of formally accepted Tuesday although multiple moments GAM at the end of the you 1st the same coefficient to GM wants Cuba and the same actually should be deleted in better in the ADM massive of system 0 2 present total energy of this instead of jail and plus 2 plants Newman of style log of some of the 2 of you as the is the is the scale to do these computations if you needed to know to the plagiarize divergent intervals and they obligated them coming in so you need to introduce the scale of the scandals not enter in physical effects you pull at the end of the scale is arbitrary OK but there is still here always at times he'd always length unified by a length it is dimension missing in the rubble and then there is a difference in numerical constantly pride and their export terms this is the 1st affects the has studied the attempts of the terrorists and enough tells that terms beyond that you can compute but but you haven't would use of Ireland Jim what of these these objects I obtained by the finite bout with respect to so To compute you need again to introduce a finite parts and an 80 continuation with a factor of a of the old to the problems OK but it has a different meaning now because the for the finite parts you needed here was because often use the divergent that it was their divergent ideologies OK while here you will have an inside divergent switches a divergent at large spatial distances but is just the same thing that you see in a different way because once you are inside the sources of their relatives of their outside you see them differently anyway deformities and you so you have this and in this thing the nice thing is that here you put the idea as computed the theory that is to say these former and the other formulas that my assistant and he raised this summer and but the number of this is an explicit it's a OK in terms of these was where intervals of up team you knew of the source of this was healing arise it so this things went into gold of something with some color scheme you knew what you do is you take said functional but you'll be placed in Union but the New New and you have the same formula also use the formula has often arise gravity that's why the formulas of an identity are useful because Jay and rise to goes to Munich where Tom you knew it is the right hand side of Einstein equations when you're writing in the form of a new people or something no I'm not sure of the signs is it's like what I was forged new new finally something like minus or plus a natural 16 by oversee 4 times maneuver when you write and sign efficiency you have this press all the nominal terms that you can put you on the loan money out of and you call the sum of the 2 new sorry that it's theirs and yours is the victim of his and so everything he added that union president on and after Nanchang accretions which argued DH promised each image in nature .period attempts you give it a name OK so right and 120 questions according the Home News but the nice thing is that the placing in this formula has 2 new new by on you gives this part of the magical moments to which you must still add another unknown term which is the editor Terry Taylor terms but at the end of this you can compute because you computing now it isn't there already is a is dominated by what happens in the source but to take stand outside the source but the the finite parts with the Alvaro Arzu will complete rounds you to define the thing uniquely and and this gives the would answer that you can compute to win the although you won't impose Newtonian theory so this game which has used a postman Koskinen and post Newtonian and the indeed something practical something which is not easy when you go beyond the leading all although but which has been pushed to quite all the and that which is the basis of what would be used the 2 computes the addition of binary black costs up to merger because you will use this posted to disposed to turn things around you to compute the waveform so the waveforms would be this HLN which was somewhere now it appeal it's a their the chairman yes it is there OK so you want to computes the limited pool of things in it that really by a binary black hole OK uh and what you do or you have this scheme which you combine sports medical scanned and posted it on and thing but which allows to compute the formed as an expansion in powers of the overseas uh this expansion will lose its value tea when you go near marriages and that we would discuss another method which is summation methods which are used in the effective 1 body formalism to go up to manager starting form of things which should not be allowed to be pushed up to marriages but didn't have to say something and soldiers 2 of electorate we can probably stop here within this new title of this I was highest at me just mentioned 1 thing that if you do this and the 1 PM approximation because this is something that is very simple in this case if you do this and the 110 approximation that we have done with before phone this formula you find and this is remarkable that the formula was offering arise gravity if you write them in terms of the contract variants components of tea which is not the same as Whiting them in terms of the core area because unknowing known and many of the exact that is to say that they believe the correct 1 TN a mass type multiple moment emitted by the source to compute the emission of a binary system of fond the must type multiple moments at the 1st posted to approximation which means going beyond lunchtime the remarkable thing is that they are given by the same formula as as the arise about it it's an accident added to pin all their now terms that are more complicated to compute that you need to tackle with that was the vandal Long Island his more good question actually what it means is that when you computes the signal which was like someone you computer new ideas because I noted that so that the technical thing is that yes I wait until it is because this thing which is good so is the sum of these and these and all the money now terms and the 1 PM happens to be the thing with the cultivar and components like that there is an acceptance because still there Monday nite is like the and understand but that X factor is 1 of the 2 Figeac plot class ,comma square on the new Conyon potential where we're not aware of any minus hope I passengers so it's so idealistic unit but it is it's a it's a lot less of a thing and when you replace the Laplacian defined does not contribute by integrating by parts because you compute when you compute the probable moment you all anything it was a lot class goes on excise exchanges 0 because it is a harmonic function so that the 110 there is something nice the number is like none nonexistent but that the segment posted on an infield post Newtonian you need to use the complicated calculations you need to use dimensional legalization but that will be discussed next time OK thank you thank them