Interview with Grant Matthews

0:00 So I'll just say that it's the 3rd of September at 10 in the morning, and I'm talking with Grant Matthews. And, well, as I mentioned, I'm essentially interested in all work that's connected with gravitational waves, especially with the idea of detecting them from real sources. and I suppose to start off with the work of yours that I've been most following of late is your work with Jim Wilson and I suppose just to kick off with I noticed and I sort of know from briefly talking to Jim Wilson at one time that the early part of this work was done back in the late 80s and so I was wondering how it kind of got started off and we started the work in 87 while Chuck Evans was still finishing his thesis he was still or he was a post I forget which he was either a postdoc or a student still of Jim's I think he was still yeah he was finishing his thesis so he was at Livermore Jim Wilson was at Livermore I was at Livermore but working in a different area but I wanted to get started in numerical relativity and some work in hydro so the three of us got together and it was really Chuck Evans that proposed that at least in my observation it may have been Jim that proposed it to Chuck and Chuck proposed it to me but suggest that we work on the binary neutron star problem actually I had two interests in it at the time one of course is the gravitational wave business but at the time there were some ideas, which are still around to some extent, that heavy element synthesis might go on in neutron star mergers, which is sort of the kind of thing I was working on at the time, which is what intrigued me about this problem. And then, well, let's continue with the history. So we worked on that through, actually, I think we probably started late 86, worked through 87, and actually had gotten quite a bit of progress on it until Supernova 87A, and then Jim and all of us were kind of diverted with doing supernova things, probably to 1987A. That kind of put it on the back burner, and we were working on it a little bit here and there, but it wasn't until 1990 then when CBRO, the Compton Gamma-ray Observatory went

2:30 up. So then there was the resurge of interest in gamma-ray bursts and the cosmological models seem to be most likely one, which then also points to binary neutron stars. So then we We picked it up again with added zeal. So that was around, yeah, early 91 or so that we really started going on it again. By that time Chuck Evans had drifted off and just wasn't interested in the problem anymore. And also early on in 87, I should mention, Steve Detweiler worked with us at Florida State. and he helped us work out the way that we deal with solving the field equations. He spent a summer sabbatical with us. But between the time when our activity on it lapsed and then picked up again after the GRO went up, Steve also lost interest in this problem. So basically it's just been Jim and I that was carrying it on since then. You know, we tried to keep Chuck and Steve apprised of what went on. So, well, so, I guess then you worked on it more or less from about then on to fairly recently when you got these results. Yeah, well, it was a long process, you know, this kind of a code, 3D Hydro, even Newtonian, takes forever to debug, and, you know, there are lots of test problems you have to run, and you get something happening, and you wonder, what does that mean? There are, you know, there are a number of things to get past, and we had a lot of trouble getting the beta equation to be stable, the way we solve the shift vectors. And so we had to develop the right numerical scheme for that. Jim also is quite a technician in hydro, and he, in fact, rewrote our advection scheme from the bottom up three times, each time going to a more modern version.

5:00 Actually, coincidentally with this project, he was working on a project in three-dimensional as a way of probing the nuclear equation of state, but it turns out a lot of the same hydro applies in that problem here. That's a special rough mystic problem for rough mystic collisions. So code developments, which Tom McAbee was the guy working mostly on that, and some other people who escaped me right now, someone named Simerman, I think. Anyways, they were sort of developing hydro in that, and as they made developments, then So the two codes were sort of developing parallel at one time, at least in the early history. Another thing that happened, when we first started writing this code, there weren't really memory, there wasn't memory available for this kind of calculation. And so I spent a long time developing a version of the code that would work with, what we had at the time with something called a solid state disk. So basically you could bring in one slice, you know, and say X, Y, one slice in Z in the computer at a time in each of the variables. And then you could do do loops over this solid state disk. So it was a disk but you could do a fast read almost like memory and a random access. But it made the coding very complicated. And then by about 1990 or so that was when we actually had enough memory we could run the code the way it should be run. And so then we had to invest a year completely rewriting it to get rid of all those. Every do loop had to be rewritten. And then getting all the, well you know how those prompts are, getting all the indices right and the boundaries There are always troubles at the boundaries. And then, well, just to come up to a more modern time, after I came to Notre Dame, a graduate student, Pedro Marinetti, joined us, and he helped develop the multipole expansion piece. Because his interest is really in gravitational radiation, that's what his thesis is on, so the aspects of this work for a particular relevance to the gravity wave signal to the

7:30 We could generate one, but this approach will be the focus of Pedro's thesis and the history. So I first became aware of the work, I think, in 1995. In any case, at the Marcelo-Bilton meeting in Stanford, I taught briefly with Jim Wilson, although I missed his talk. My impression was that at that time you hadn't quite got the final results, but it was soon after that. At that time, we had our first, first, well we had results, I gave a talk on this already in 1988, we had our first results in the sense of, you know, we could put a neutron star on, we could solve field equations, but we didn't really have the shift vectors worked out right until then. Was that 94 or 95? It must have been 94. Because it was just before I came out here, which was 94. And that was with the gamma law equation of state. Let's see. At that time, yeah, we were still puzzling over what was the building, what wasn't, what was the code doing. So in fact, for that meeting, the data we presented were a calculation of two stars about 0.8 solar masses. So they were almost Newtonian. I mean, they were not strongly relativistic because we trusted the code with that. Yeah, I can't remember what the other details are. But the other thing was, as everybody is and has been in the field interested in we were interested in the final moments of merger and at that time already we were we had noticed that it was hard to bring the stars in depending on the equation state and and keep them stable so we're by running low mass stars we could actually do a merger and that's still true if you do a low mass star or a equation state, then you can bring the starts in to merge. And so we showed data then, but we didn't publish it in the paper of the actual merger in Jim's talk.

10:00 Yeah, I see. I missed Jim's talk, but I gave a talk at that meeting based on work that we were doing in our group, which was more or less, well, here's what we can tell so far about how you would go about detecting merger signals, except, you know, nobody has done anything much except Newtonian or the post-Newtonian, so what we really need is some fully relativistic code, and so, of course, he came up afterwards and told me about what you were working on. Oh, okay. Oh, so you met Jim, then. Yeah. Good. Yeah. So, at that point, you already, I think you were saying, you already had the situation that with the... Well, I remember the runs, you know, it's a little vague in my mind, because it wasn't until, you know, early in 95, well, late 94, you know, early 95, when we were sure we had all the bugs out of the code. But I do remember, yeah, Jim's decision at that time, we were running these gamma law equations of state, and we'd set up what we thought was a reasonable initial condition and the stars just didn't act like we thought they should my memory is they would even expand or contract but whatever it was they were doing decided that maybe this was an artifact of the gamma law we were using that's right because we were using gamma laws are tricky you know in order to get if you want to have a 1.45 if you want to pin a mass to it and pick a gamma equal 2 you either have a very fluffy star with a radius of 15 or 16 kilometers or else you have a compact star but it's right next to being able to collapse I remember this now in this discussion we're going through it again because one of the papers that's shown up is based on a gamma law which we tried to do this problem again. And if the star is very fluffy, then as they get close, then there are big tidal effects and they start taking a storm. And if the star is too close to the maximum mass, then of course, we think we understand a little better what's going on, but we were having a hard time getting a stable star. So the decision we made then was to go to a realistic neutron star equation of state so that we didn't have to deal with

12:30 And then, after we made that, and then, of course, did a series of, you know, hydro tests and so forth, started running and looked for everything we could, we see the stars were still not stable then. That was our first indication of something that was going on. So how surprised were you by the, by seeing this situation where the, where the stars themselves were unstable as they approached each other? Well, I mean, nobody anticipated it, right? Right, right, sure. And, yeah, I mean, there's a lot in the motion of the stars and what the stars are doing. I mean, it's not just that they're unstable. There are these peculiar vortices that show up in their fluid motions. It's a, it's a, but I'm not answering your question. The question is how surprised were we? All I remember is, it was after I'd come here, we were working on our paper, they were just describing how you do the model. We were discussing the calculation results and we were, we were going over it. I was at Livermore talking to Jim, and we went through the different pieces of it, and Jim convinced me that it was real, and there wasn't anything spurious that we could think of. And then I said to Jim, well, we're going to have to write a Bizarre letter, because if this is right, then this is a new instability. So I don't know how surprised we were. The hard thing is, in a big numerical code, is to believe it. And then once you believe it, then it's not a surprise. But we certainly didn't expect... What we had expected is what everyone else has always proposed, that, well, if the stars... Well, we started them out as co-rotating, so they should more or less stay co-rotating, and they would just tidally distort. merge. That was what we expected. And instead what we see is that they develop these peculiar

15:00 fluid motions. So if I get it right from what you were saying, at first it was sort of a period where you were seeing basically this effect for sort of certain extreme cases and go away from that and look at other cases and then gradually became clear. I think we were seeing them in the gamma equal two equation of state, which is sort of the standard thing everybody does for a neutron star. It's a kind of a stiff equation of state, but it makes the stars sort of fluffy. But it's a simple thing to do if you don't want to put in any nuclear physics. Now, Jim could give you a better perspective on that. because he was doing most of the numerical runs that time. And I know we spent at least two years, I would say, tracking down aspects of this problem that we were probably seeing at, well, at least a year. And, you know, finally, you know, there are different ways that it shows up. I mean, you start just seeing that alpha goes down, you know, approaches zero, elapses, goes zero, and phi squared goes up. and also we were having this troubling thing that the beta was not very stable so we were tracking down those problems all the time and one of the things we found at first since then I've found that people knew this all along in fact it's in one of York's papers if you try freezing the matter distribution and solving the fields there's an instability that comes in you in fact have to let the matter respond to the field And that was one of the early things we ran into. And then we found numerically that we had to let the two go hand in hand. That was one instability, but that's not this one. I can't say exactly when it was, but I do know, and I did runs myself with gamma laws.

17:30 For one, it was very hard to get the initial condition for something that would actually work. And we could find a star that was stable, but it tended to be a low-mass star. But once again, to reiterate, it wasn't until we went to a realistic neutron star equation that we were convinced it wasn't an artifact of just having these fluffy stars, which a camelot gives. It's either fluffy or it's too close to the maximum mass. I forgot what the last question was, so I probably didn't quite answer it. Well, I think I was basically getting at just the idea of... I mean, I'm generally curious, especially, as I say, from my own experience, which is not so heavily numerical, but I'm certainly familiar with the problems of using code and when you get to the point that you trust it. just interested about the general idea of when you get to a point where you feel, okay, so now even the results that I wasn't expecting, I kind of believe because I think that the code is not pathological anymore. Yeah. Well, I mean, there are standard tests. You know, you put one star on the grid and make sure it sits there. You know, we know the POV solution, right? The Schwarzschild solution. And then there's an analogous solution, which you have to find numerically coordinates like we use but it gets the same mass so you know you have to get the mass right and I mean so those are tests I mean if there's some hydrant instability it shows up there well but you I mean you know you there's something called the Riemann shock tube problem there's an analytic solution for just a linear shock with certain boundary conditions two zones of pressure equilibrium, two different densities rarefaction wave and so forth going back and forth you can see how well the hydro is behaving by running that in different orientations so it's a good test I mean there are a whole bunch of things like this that you do just to test the hydro piece yeah, well the GR piece you sort of test with a single star

20:00 Let's see, there are some other test problems, which I don't remember right now. Sure. Yeah. Okay. I mean, so, I mean, there, again, Jim mostly did that. I mean, there's one thing where you just, when you set a star, there are some oscillations. You can look at the oscillations. We did that early on. Of course, every time we rewrote the hydro, then you had to do a series of tests over again. then there's a single rotating star now that's one that we did some but haven't done exhaustively we should probably redo because others have looked at that problem we should compare it with that there is a test problem that we're setting up to do now which is Chuck Evans for his thesis he did the axisymmetric collapse in exact relativity from. And so we can run it in our code and compare. One of the things I'm trying to get Chuck to research in, in fact, is whether or not the density increased in those simulations as the stars came in. I think one ought to see something similar, but I'm not sure. So But it depends upon how the stars respond. Well, then I was curious, I suppose, as to what was the kind of, for instance, when you published this red letter and spoke at conferences, I was curious as to what kind of reaction you encountered. I mean, I was interested in the, I suppose, part of the problem of extraction of signals was from finding neutron stars, or at least I was just sort of leaving the problem actually as much of this happened because I was writing my thesis which was on other things. But of course all along we had just been completely expecting the neutron stars to just happily come in and kind of merge and of course we did find that things were different from what we expected, even on the basis of the existing Newtonian and post-Newtonian codes that we looked at. But I can even remember giving a talk to some people and somebody said, okay, so, you know, doesn't this turn into a black hole? I was kind of completely on plus. I said, well, yeah, I guess after they merge, it's going to turn into a black hole,

22:30 but I'm not even interested in that. And I thought it was curious because then this result came out and I thought, well, I'll show you how much I was thinking about that possibility. So I was curious as to what was the reaction that you generally had. Well, you know, I think the field has responded with healthy skepticism, which they should do until somebody reproduces it. Now, today, no one has done a calculation that would attempt to reproduce it. People have done simpler calculations, you know, multiple expansion, a linear order, or a first post-Newtonian, neither of which should show the fact that neither of which do, so that doesn't really say anything. people the most serious calculation was the one that the NCSA and Cornell people did where they imposed a killing vector enforce rigid co-rotation that seems to damp out the effect and in fact right now I'm running a series of calculations with our code where we have also damped out rotation and indeed i'm seeing what they see although i haven't been able to match i mean i haven't attempted to match exactly their calculation but i think qualitatively i'm getting the same thing which is that the frequency goes up and the density goes up it goes quite actually it goes up quite a bit and then you see large tidal distortions so qualitatively i think i'm seeing like what what those people see um so you know people are sort of looking at pieces around the problem but they're not looking at the source terms that we think are driving the effect um until someone else develops a numerical code and i'm hoping the grand challenge people will do that and actually let the hydro respond I don't think anyone is going to be convinced no matter what we say but okay so back to your question how did the community respond the people that didn't have that had a curiosity in the field but weren't actively interested in it I think we're all very very interested in especially the gamma ray verse because this this opens up a new possible paradigm for Gamma Ray Burse. So we had a lot of positive

25:00 feedback from that community. At least the people I talked to. but the let's see, the people with a vested interest you know their ambitions are very skeptical. You know probably for a good reason, that this is something new and unanticipated. Let's see. So, a lot of people have the immediate response that somehow this is a violation of the principle of the quote-units. And that has, there's sort of an ingrained intuition that people have about the frame of the star, that it's moving along geodesic, and we sort of naturally think of that killing vector kind of rigid co-rotation frame where not much happens. And it's just a peculiar thing that there's this extra term, which we've identified in our papers, which is sort of this velocity dependent. It comes from, you know, the covariate derivative. There's this, you know, the Christoffel piece of that covariate derivative. When you write out There's this extra piece that looks like a ui, uj times derivative of the three metric. That little piece is not Newtonian. It's somehow the motion of the star against the background curvature. And the star, the fluid, can respond to that. you know it so if you impose rigid co-rotation and then you set those used zero you can pick up a frame where velocity goes to zero right but um if the if the star can just sit there and say well you know you know how does the you know the fluid can respond to that there's a force term there you know and it no matter how it starts out including co-rotation it finds out well it can start

27:30 relaxing which actually involves a fluid motion and and set up you know there's there's a force and an energy associated with that and actually increases the gravitational binding energy by responding to that that little piece and it's not intuitive you know there isn't the newtonian piece i think there's a start of it you can see even at first post newtonian order but that caused at the first post-Newtonian correction of gravitational potential. That caused some confusion in a couple of papers that have come out because it's not a first post-Newtonian term. It's like a fourth order term. Let's see. I keep going to talk about the physics and you want to ask about sociology. No, I'm interested in physics too since that's what drives me. Yeah. For some reason, well, You know, okay, but, okay, the sociology of that. I mean, we added an appendix because, let me say, let's see, our second paper, our third paper, it was an FJ paper, we had a referee that just did not want to see this paper published. There was no science in the criticism. It was just, in my opinion, you know, I've never had a referee report on this that said, let's see, the only way that I could check this to do the calculation myself, and therefore I can't accept it, you know. Now, you know, the guy wants to check it himself, he should check it himself, but you know, he ought to at least give us the benefit of the doubt that we checked our code. I mean, there have been some papers that came out that said, well, maybe the boundary conditions are wrong or something like that, and we checked all the boundary conditions. But let's see, of course we're always looking for something new and we're always looking Let's see, I want to get in, let's see, we were talking about this term. Oh, yeah, so the referee report, and he said, well, you know, unless you can see it in some kind of post-Newtonian expansion, you know, if this is real, it ought to show up in some kind of post-Newtonian expansion, you know, in some order. And, you know, why hasn't anybody seen it? So, you know, we add an appendix to try to appease the referee where we just wrote out the equations of post-Newtonian at, you know, at the, let's see, the second order correction.

30:00 You know, the first correction to the Newtonian potential, there's a velocity dependent piece that comes into the potential. Now, then if you take a derivative, you know, although I don't think it's exactly the term that's driving this because it's just the mass energy associated with the motion, but if you take the derivative, you know, for the gravitational force that includes that velocity-dependent piece, you'd have something like what we have going on, and there's a stronger gravitational force. So we tried to point that out in the appendix. And now, if you take the derivative of that term, then it's higher than first post-Newtonian. But sure enough, somebody wrote a paper saying, okay, these people, you know, the crazy sentences are like, well, I would be inclined not to believe this whole thing at all, but these people say it shows up at first post-Newtonian, so we have to take it seriously, which is not what we said. And then the paper goes through and arrives at first post-Newtonian because it's higher than the first post-Newtonian, so they don't see any effect. And I wrote them, and I said, well, look, you should publish this saying. Well, this is a demonstration that what we had said was correct, that it wouldn't show up at this order and shouldn't show up. But there's a communication problem or something. So that, I shouldn't complain about what other people are doing. But, uh, anyways, uh, we also, well, anyways, we tried to explain to the authors that that was a mistake. That you're sure that that was not a mistake? Yeah, uh, but, uh. But, uh, I'm not sure, um, I know Alan Wise, but. Yeah, that's the paper, that's one of the papers. The same mistake occurs in the Brady Hughes paper, where they did a multiple expansion instead of post-Newtonian, but only retained linear order. And so you can see right away that the terms we would say is causiness isn't there. And then Allen tried to identify a completely unrelated term that has nothing to do with

32:30 what we said in our paper. there's a sentence in the paper this is unambiguously he's unambiguously identified the source of this compression effect and it's absolutely not true and I told him that I don't want to complain about Alan I mean you know he's he understands post-Newtonian and he's doing what he understands but I told him then and I've told him since in fact I've even tried to get him together with us, that he should do a second post-Newtonian, or at least something that has a term of order, you know, M over R, V squared. That's the kind of term we see. That connected with the effect. But you were saying that most likely you think that the real sort of star-crushing effect comes in at fourth post-Newtonian order? Well, not fourth post-Newtonian order. It's a V to the fourth. Okay, so, it's not Newtonian plus corrections of order V squared, or corrections of order M over R. You have to go to the next order. I'm right, well, there's actually, the reason for people having a hard time, you know, Jim Wilson, he could give you a perspective on this too, he talked to him, but he talked to Jim York about this, the way we were solving the problem a long time ago. And what Jim York predicted turned out to be true. He said, what we're doing is probably going to be right, but nobody's going to believe it. And there are sort of two things in this. One of them I think is kind of a red herring that throws people, I think, off the track. Just, you know, the way we solve the metric is this thing called the conformally flat approximation, where we force the three metric to be, you know, a flat three metric times a conformal factor. Now, that simplifies the equations. And it's actually the standard approach to the initial value problem for two black holes or two neutron stars. But you know, and in fact,

35:00 we solve exactly the constraint equations of relativity. So in fact, it is a solution to Einstein equations. But what you're not sure of is exactly what the physical situation is that you solve. You know, no one has ever, you know, there's a way to prove that a conformally flat solution exists. There's this thing called a cotton-york tensor, or there's a piece of the wild tensor, that you do, I think they're the same thing, that has to vanish in order for, to prove that, you know, a conforming flat solution exists. I don't think anyone's ever done that for, you know, say, two neutron stars. I mean, you can show it in some simple geometries like spherical or Robertson-Walker or a time-symmetric, like colliding gravity waves. I mean, there are a number of those proofs that exist, but they're cases with a high degree of symmetry. So what we're doing is, you know, is solving a series of initial value problems using that conformally flat approximation. In essence, when you use conformal flatness, you've imposed some constraint on the system and you don't know what it is. Let me say it another way. the axisymmetric calculations right, they put two stars on the grid, you know, say they're colliding, and then they start evolving the evolution equation. So although they started with a conformally flat metric, it's, you know, their gamma dot, you know about the ADM book, so, you know, gamma dot or k dot terms start to develop, and so the off-diagonal terms develop. And what they see right away, after they start the evolution equations conditions is that a gravity wave develops and flows off the grid. So there was hidden gravitational radiation in that solution. You opposed it conformally flat so you cannot see. There's no transverse, traceless part of gamma. So you don't see any gravity waves. But they're in there. So you don't know how much your physical picture is distorted by the fact that there's this hidden radiation. Now, we can do some estimates because we can estimate, say,

37:30 the mass energy in the gravitational radiation from something that York worked out a long time ago. You can split the extrinsic curvature into a transverse and a longitudinal piece. And we can calculate the longitudinal piece and we use a constraint equation to get k and then we can see what the residual is and it's a small piece of the problem. So we think that it's probably quite an accurate description of the two neutron stars that hidden gravitational radiation is a small piece of the problem. But it's kind of a red herring because people see that and say, oh, well, they've done something with relativity. So that is somehow influencing their hydro and what they're doing. And so the fact that the stars compress must be an artifact of the conformally flat approximation. That's sort of the easy way out if you want out of it. You know, I can't find a measure of what imposing that approximation does that would justify a statement like that. But of course, until somebody actually does the evolution equations, we won't know. But just to summarize what I was trying to say, is that one of the reasons for skepticism is because we impose this conformally flat condition. it raises suspicion. Now, back to the gravity wave problem, it is a problem there because when you impose conformal flatness, you've actually kind of picked a kind of coordinates or a kind of initial condition where in one sense you could think of it as that you imposed incoming gravitational radiation exactly canceling the outgoing radiation. That's why you don't see it. I mean, there would be an electromagnetic analog in that. Yeah, sure. So in a sense, you've chosen some special coordinates and special initial conditions. Now we do another orbit. We solve the initial value problem again, say the stars are closer in. Now we've actually imposed a different condition on that binary system because now there's a different wave coming in, canceling the outgoing wave, or whatever it is, radiation is different. Now, since that's a small piece of the problem, our view of it is we can

40:00 connect those two approximately by using a multiple expansion and saying, well, this is how much gravitational radiation would have gone off between that orbit and that orbit. And then we can reconstruct the time sequence and therefore the gravity waveform. But it's definitely an approximation because in a sense we're connecting a bunch of initial value solutions which are not physical initial value solutions there there's some approximation to it which is you have a measure of but we don't know it exactly so there's definitely going to be an approximation of time history that comes in that way both from the fact that the multiple expansion is approximate and from the fact that we're connecting to initial value solutions that are not the sequence of physical states. I was curious along those lines, is there any sense in which, I take it there isn't, but I'm sort of used to doing problems like what I'll be talking about later on, where you impose an adiabatic condition and you assume that a particle is in orbit in a geodesic orbit and that the radiation is just moving it from one geodesic orbit to another. So I was wondering, is there any sense in which there's sort of an adiaticity condition here? Well, that's what we were... I mean, to reconstruct the waveform, that's exactly what we do. I mean, we essentially, we turn off the radiation, you know, the back reaction in the evolution equations, So we get stable Keplerian orbits for two neutron stars, and then we increase the angular momentum or decrease the angular momentum, and then find the new stable Keplerian orbit until there's no stable orbit. So, and then, yeah, to connect them, we do just what you said. We say, well, we can use a multiple expansion and guess how much radiation would have moved the orbits in and out in. Does that impose, in your case, any condition on the rate of inspiral, as it were? I mean, if the rate of N-spiral is too great, does that kind of prevent you from finding a stable non-radiating orbit?

42:30 I mean, it can... Well, it means, yeah, at the point when the gravitational radiation loss is a significant fraction of the energy in orbit, then, yeah, this is not a good approximation. The same as in the case of what you're talking about. So, about where does that point occur within the two neutrons? Well, what we find is that the orbits become unstable when the stars are still a long ways away. So, actually, the gravitational radiation has not caused the orbits to become unstable yet. It's about a part in 10 to the third. I think J dot over J is like 10 to the minus 3. So, basically, you don't ever run up against the problem simply because of the dynamic instability? This is right, yeah. This other peculiar, it's a relativistic instability, but it's an instability in, if you like, the gravitational potential, you know, the relativistic potential. It's not radiation instability. Now, in terms of this potential instability sort of blasted sort of orbit or whatever we want to call it, well, my sense is there seem to be a huge range of different estimates as to whereabouts this takes place. But do you have any sense that there's any convergence in the different estimates as to where it's going to be? Yeah, well, okay. If you have two points, there's a correction for just finite extended stars. You know, obviously the potential is different. You know, the tidal forces are different. And so there have been, you know, the old calculation, you know, know, Kidder, Will, Wiseman, that sort of thing, were two points. And then the inner stable orbit moves out a little bit if you go to extended stars, and that's what last year Shibata and I think Fly and Wiseman both had papers on that. Now, if you add in the

45:00 effect we get, then it moves out again, which is this other term that drives a deeper potential. So, to me, they're all consistent. Until somebody adds the physics that we see, they And they won't agree. Just to return to something you said earlier about the attitude of the theorists, I suppose, whose particular interest is in gravitational waves and so on, and you mentioned...