Interview with Luc Blanchet

0:00 It's the 12th of October, around nearly 3 o'clock in the afternoon, and I'm talking with Luc Blanchet. So, you were saying, Luc, about the work of Birkinthorne. Yes, for me, I think that it's my own point of view that the modern history of gravitational radiation reaction began with the work by Burke and Thorne in 1969 because they found an expression for the radiation reaction force, which is extremely simple and which was at that time justified by the fact that it gives straightforwardly the correct amount for the total power extracted by this force from the source, which is in agreement with the standard quadruple formula. At that time, however, the proof of the validity of this force was not completed because it was only done within the linear theory while one needs, at least in this particular coordinate system, they used at least one iteration of the field equation is needed. So there are terms coming from the quadratic order which are needed, in fact. But if you include these nonlinear terms, then you can see that the end result is exactly the one of Perkinson, so you can prove it, but if the proof has to be consistent, you you need to include them. Because, in fact, if we want to see exactly why this is not consistent, I don't know if you are interested in detailed technical points, but they start, for instance, in the Mr. Sondwiller book, they start from the expression of the resistive terms at 2.5 post-Nitian order, in harmonic coordinates. So there is a contribution in the G0,0 part of the metric,

2:30 as well as the G0,I, and G,I,I, and G parts of the metric. And if you consider only these terms, and compute the power associated with the radiation reaction force, you see that it does not agree with the Einstein-Cortopol formula. so there is a need to include non-linear terms so that they agree so in principle you cannot start from this linearized metric in harmonic coordinates to compute to make a linearized coordinate transformation and get only the Bergstorne scalar to derive them but it's I mean this has been proved by other people an interesting point is that expression for radiation reaction force depends on the coordinate system the standard property of the gravitational field this is due to the equivalence principle so this, the Bergthorn expression for instance does not agree with an expression formed by Chandra Sekhar and Esposito of the, I think it's 71 or 70, maybe 1970, where they worked with at the second and a half post-Newtonian approximation and in their work they obtained by means of a direct post-Newtonian expansion previous work dealt with the first post-Newtonian approximation and then second post-Newtonian approximation by Chandrasekhar and Nupku, which was very intricate, in fact. And the work by Chandrasekhar and Esplito formed an expression for the radiation reaction which was completely different from that of Birkensohn, but was later proved to reduce to Birkensohn by a coordinate transformation by, I think that this is a paper by Boni, maybe. I can give you the correct reference if you stop the equivalence between different forms of the radiation reaction depending on coordinate systems there are many of them in fact for instance there is a short letter by Gerhard Schaeffer

5:00 who used who made a study of different forms It's, in a sense, trivial that you can add to the force can be expressed in the form of derivative spatial and time derivatives of metric coefficients. And metric coefficients where the freedom of gauge in them, so there is a freedom in gauge of the force, which is not in gauge invariant, so there is an infinite number of equivalent forms. I think that maybe Thibault yesterday probably spoke to you on the form he has used for the binary pulsar, which is also still another form. When you compute its total power and also angular momentum and so on, it agrees with standard formulas calculated from fluxes at infinity of the source. After these works, well, after the discovery of the binary pulsar, there has been a big controversy concerning the applicability of the Einstein quadrupole formula to the binary pulsar. I think I will not enter this controversy. I think you have many informations on it concerning the... I think it started with a paper by Jürgen Ehlers and the people Rosenblum, Avast and Goldberg, I think. Walker and Will also studied many papers calculating radiation reaction forces and pointed out that the only one among these authors who get correct expressions are the ones who have made the correct iterations of the field equations up to the third order generally. But this controversy now has ended up, and it can be said that the quadruple formula is completely approved even for systems containing compact objects by the binary And now it is interesting to ask what are the correcting terms to the leading order Berkshorn expression

7:30 that we can take now as completely correct and we can also use this expression But it is only the dominant term of many other terms, so maybe I can discuss the work I have done on that. first of all first of all the Bergson formula is valid for a system made of some fluid and this system is in slow motion because the Bergson formula is a leading term of that it is valid also when the first radiative moment is a quadruple one. But Burke Thorne, in his original paper in the Journal of Mathematical Physics, has given an expression for any health order multiple moments which are derived within the linear theory and which are in fact not correct if the system has all its moments radiating. More precisely, the Burke-Thorn formula is only for a scalar. It is only a scalar potential whose gradient gives radiation reaction force. But if you assume that the first radiative moments are formed on the octupole, not the quadrupole, then the formula is no longer correct in the sense that there is, besides the scalar of Birk and Thorne there is also a vector contribution which is in fact an analog of magnetic type or current type moments and for instance if you would compute

10:00 the radiation reaction force from a system radiating moment, that is non-static moment, is the health order moment, then there are two terms in fact in the radiation reaction force. And this is, I think, this has for instance to be included in the works by people who study the stability of neutron stars against radiation reaction. the Chandrasekhar-Flickman shoots instability. People generally compute the L equal to 2 modes, L equal to 3, L equal to 4, and so on modes. However, in computing higher order modes, you need to include also current contributions to the radiation reaction force, which is in general not taken into account. And I think that a lot of published works on that are inconsistent, are only consistent for the L equals 2 modes for the stability. So we have done that in a paper with Thibault in physics. I can give you which you wish. So this is if you include... So at the lowest order, you recover the usual Bergthorn potential because the current spin contribution is constant, so it does not radiate. So it's only the first correction, the first difference is at the L equal 2 level in the current moment, which represents a contribution of the same order as the L equal 3 contribution to the scale A and to the scale A. I see. So, you've evaluated this particular correction to this particular order. I'm not really sure myself. In a particular system, such as the binary neutron star, how small is this correction? Oh, it's of first post-tuntonian order with respect to the dominant radiation reaction force.

12:30 That is, it will be something like, I would say, 1 over C6 in the equation of motion, instead of 1 over C5. No, sorry, 1 over C7 in the equation of motion instead of 1 over C5. That is, it is one post-mountainian order below the first radiation. That's interesting. people yesterday mentioned this work. This work, of course, is very recent, but maybe we should go back a bit and you can explain how you first came to work radiation reaction, I suppose, but gravitational radiation in general. I suppose your own personal history in the field. On the problem of gravitational radiation in general, or for radiation reaction effects? Well, I suppose in general, the gravitational radiation in general. Well, that's very simple because I started my thesis on this subject and my advisor was Thibault and my subject was to investigate the non-linear structure of the gravitational field in the the vacuum exterior region of an isolated system by means of of vertical art series. And so we I started in fact from the from the Kip Thorne paper in Review of Modern Physics 1980 where he gives So when you first began as a graduate student, for instance, to work on this problem, were you aware at that point of the mildly controversial history of this? Oh, I became aware of that, I think, yes, rather happily, yeah.

15:00 At that time, Thibaut was working on the binary pulsar most of the time, so I knew about this controversy. I knew, for instance, that there was a need to complete the equations of motion of two stars and including all orders up to the radiation reaction order to check that. When did you begin your graduate work? I think in 1982. Now, at that point, the data from the binary pulsar on radiation reaction itself, that is, data concerning the loss of momentum in the binary pulsar had already been published for, I guess, for two or three years. Oh, yes, yes. So at that point, you probably maybe don't have a feel, since you might not have been familiar with it before, did you get the impression that there was still a current controversy over them? No, I don't think so. No, no, no. I think that it was already clear in the beginning of the 80s that the quadruple formula could be applied to such systems. Sure. So most people that you were familiar with were happy with the experiment then? In fact, I think that Thiebaud probably had completed his work at that time, so I think that it was pretentious, that it would apply. and I suppose in fact Thiebel himself mentioned of course that in a sense from the point of view of the formula itself it's only now for instance that you and he have produced a qualitative analysis of the next leading order term as it were that one has an actual you could say that one has an actual definitive proof exactly to what extent the cluster perform is.

17:30 Yes, exactly. In fact, you can do that in two ways. The first way is to compute explicitly the corrections to the radiation reaction force. It was not exactly done in our letter of 84 because it was only valid for systems whose lowest moments are static. But for a system whose lowest moments are radiating, especially the quadrupole, of course, then the next order term is more difficult to get. and in fact this has been completed in a more recent work of 83 by by me using using a general method of matching or in fact a general iteration of the of the the external field And simultaneously with my work, Cliff Will and Ballard Ayer published a letter with the next order radiation reaction force valid for binary systems. And for some time there were disagreements between both expressions, but it was not easy to compare between general formulas that I have, but these formulas extend the Bergson formula in the same type of coordinate system, and formulas which are valid only for circular orbits but for in a general gauge. And in fact, this is now shown to be consistent. So we shall clarify this in a common paper with Clifford-Will and Ballarier. So it can be said that now radiation reaction force is complete half post-Newtonian order. You see, the first order is 2.5, 5 half, and then the next order is 7 half post-Newtonian order. And furthermore, it is in fact complete up to the next order

20:00 also, which is 4 post-Newtonian order, where there are tails which arise. And this, we in 1988, with Thibaut, in physical work. So you were saying that when comparing both of these worksheets, your own and Will, that initially there was a problem that it was difficult actually to decide whether they agreed or not. Yes, yes. But now it agrees perfectly. However, there are subtleties, for instance, the Bergthorne radiation reaction force, which is the lowest order, then modifies also the Newtonian potential generated by the source, because the density, the mass density, is modified by a small resistive contribution. So this modified potential will give a contribution which is also time-odd or resistive and this will also enter the equations at the 7.5 post-Nitian order exactly at the same level as the other one So this is only when you include all modifications all time-odd modifications in the equations that you can get agreement So it's not so easy, but it agrees. Interesting. I'm curious, just thinking about it now, I don't seem to have any sense about this, but are there any physical systems that you could conceive of where the leading order radiation reaction term would be suppressed sufficiently Yes, I think that Bob Wagoner investigated such a system at some time. It is also possible that in perturbations of neutron stars, you can do such an assumption

22:30 since you are considering a particular mode of vibration. And so if you consider, for instance, the very high mode, then the contribution of lower order moments will be suppressed because they do not react to such... Well, this mode does not induce such moments if they are... So maybe in these situations you can assume this. Yes. But I can maybe recover for you the Wagoner paper where it is mentioned some physical situations. I don't know if they have any relevance. I'd be interested to see since I haven't seen the paper last time. Well, Ben, this is sort of a more speculative question, but do you have any real hope at all that this then is an even smaller thing than the quadruple formula thing? Do you have any hope at all that it might be detected experimentally? Can you imagine where the leading order term was beyond the quadruple term, which might be detectable, for instance, by the computationally detectable? For coalescing binaries, it will surely be detected in coalescing binaries. Do you think that in the coalescing binary that, say, LIGO or VIRGO would be able to pick out the... Yes, certainly. In fact, as soon as LIGO and Virgo can detect higher-order corrections in the radiation beyond the quadrupole waveform, and then if you can detect a higher-order term in that, This is equivalent to detect an higher-order term in the radiation reaction force beyond, for instance, the quadruple term, the linear term. So you think that the second-order term, for instance, in the radiation reaction would provide some sufficient modulation of the wave form? Yes, certainly. But in fact, this is up to a certain approximation. It is better to work directly with the energy, the total energy radiated in the wave, than to work on the radiation force acting on the orbits of the binary.

25:00 It's easier, in fact, to compute the total energy, and the Caltech group, for instance, has computed, has included post-Syntonian corrections, and also a three-half post-Syntonian correction in that, and these are very important, and in fact, what is needed now is to go much much beyond these orders, in fact. But for the binary pulsar, we have proved that the next order corrections in the radiation, or equivalently in the radiation reaction forces, cannot be observed. For instance, if you consider the formula for p dot, the orbital p dot of the binary pulsar, It is, numerically, it is approximately 10 to the minus 12, or minus 2.4, 10 to the minus 12. And the correction to this formula, including the first post-Mittonian correction in the radiation reaction force, turned out to be 10 to the minus 5, smaller than that, for the binary gulsar. So it cannot be detected. And we have also computed what are the contributions due to the tails in that, which is 3.5% beyond that, and this is 10 to the minus 7 of that. So there is no hope to detect that in the data of the binary result. But in coalescing binaries, it will be detectable. And in fact, even more, there are effects which will be detectable at a very small order. Lower orders again, perhaps. I see. Well, to get back briefly, if there's anything more to say about the...

27:30 that you can think of between the period that we were just speaking of when you first began to work in this field and your current work with Hevo on the next order effect in radiation reaction were there any were there any significant issues in the interim period in the late 80s I suppose that you recall that were of importance in the field? Well, in view of coalescing binaries and the detection by LIGO-Virco, it will become necessary to know exactly the radiation reaction force acting on the orbit, and not to only compute what is the average energy flux emitted by these systems. I think that if you want a complete treatment of that, you should... I think that at a level like the third post-Newtonian order in the radiation field, beyond the quadrupole order, which corresponds in the equation of motion to an incredibly small order, because it will be the third post-Nitian order plus 2.5 post-Nitian order corresponding to the radiation-rigation force, which is 5.5 that is 1 over c11 in the equation of motion to go at this order which probably will be needed in these systems since you can observe the radiation with an incredible precision you have a large number of cycles that you will observe and I think that at this point it will be necessary to control the radiation reaction force on the orbit rather than the total energy flux to get the correct one. I don't know if it is possible. That's right.

30:00 Well, of course, this is an issue since there's obviously a monumental calculation or task involved there, which you're in the forefront of, I mentioned. What sort of time frame would it be possible to complete that project? I think that within one year we could have the 2.5 post-continentian order in the radiation field. the second post-continent order is completed and the next order will be done I think that within one year it will be possible but you see at this order it's somewhat tricky because at this order it is 2.5 post-continent order that is 1 over C5 in the radiation in the waveform That means that you need the equation of motion also to compute these waveforms at 2.5 order. And so these equations need to include also the radiation reaction terms themselves. So it's a little tricky. in principle if you want to think only in terms of radiation reaction forces acting on the orbit or if you think only in terms of equations of motion the order you should go is extremely high because it is the 1 over C5 or 2.5 post-Newtonian to the leading order radiation reaction force, plus all the corrections you need in the radiation, this one. So if you want to go to 3pN order in the radiation, you should add to that 2.5pN order to add the level and the equations. But in fact, to get also the expression of the waveform,

32:30 you need also to include, I mean, to reduce the expression of the waveform, you mean there are also the radiation reaction effects to be included also in that. So it's a little tricky. I don't know if I am being very clear on that. Well, I think I've got the answer. In fact, why don't we...