Interview with Thomas Baumgarte & Masaru Shibata

0:00 Thomas Baumgartner, 16th of March, 1999. Put it down for prosperity, or at least for me. As I should mention, actually, although we're past controversial studies, in addition to clearing with you, if I want to quote you, no one else will listen to this, unless I get it over, unless you say so. Okay, so, well, so maybe you'd sort of go over a bit. Well, you mentioned, I guess, so I was interested being at the Austin meeting because it was clear that just stability of codes is a real bugbear in this field and that to have a code such as you were talking about that actually just runs indefinitely, I guess, is a big breakthrough. So maybe you could explain a little bit more about the background to that and how your discovery came about. Okay, so really, okay, so this started maybe a year and a half ago, okay, when I came back from one of the other meetings, and we, or I decided, and Stu supported me, okay, and basically he told me, well, we should really get started and get our own 3D code, but he left it completely open to me to pick what I wanted to do. and at that point there were three options of what we could do and basically the idea is how could we get interesting results and at that point we were looking at three different options one would be code basically based on this Wilson-Matthews approach where basically you neglect all the dynamical degrees of freedom and you do this conformal approximation thing and I was very reluctant to do that because I don't while I that is a good idea, and it's a nice thing to do. I don't think that the community has a lot of respect for that approximation, so I think basically there's not a lot to be won, or not a lot to be gained, but it doesn't necessarily reflect my own view. The other possibility that I think would have been very interesting is a perturbation approach, where Basically, you write the equations as a background, plus perturbations, but you don't pick the background a priori.

2:30 Basically, you just write it as, basically, for example, Gij0 plus delta Gij, or something like that. And then you can write that formally as a talex, or as an expansion, and then you can drop all the terms higher than linear in delta Gij. then you basically you can write a very general code and then in principle you can insert some background solutions which you already know and compute the perturbations on top of that and that's interesting because for example for the binary neutron star problem because we have background solutions we basically know the quasi-equilibrium solutions so it would have been interesting to actually evolve them or basically insert them as background top of that. So I think that would have been an interesting approach. However, of course, it's still limited because you can only do so much. You can only basically look at situations where you already have a background solution and so forth. And then the third possibility was a fully self-consistent 3D code. basically I thought that would be the most interesting thing to do. Also, a little earlier actually, Masaru Shibata had visited us and he told us a little bit about the code that they had developed in Japan, and he, what I still remembered from that is that he said that if you introduce this new basic vector, even though it's not really a vector, but these, you know, vector-valued auxiliary variables, that you can dramatically improve the stability of the systems. So I went back to a publication of theirs, and in which actually it's not clear at all if it's stabilized the system show very early evolution, and they don't discuss it, they don't compare or anything. So I went back to there, and indeed, I see they introduced this vector value, three vector value set of auxiliary variables, and they do their conformal decomposition and everything. And so that got me started thinking about something in that. So I basically realized, well, maybe if I introduce auxiliary variables, maybe I can improve the system. Now, the story is that the ADM system, at least coupled to outgoing wave boundary conditions or something that produces insubilities,

5:00 or basically the folklore, the heuristic argument, is that the Ricci tensor is not hyperbolic. No, the three Ricci tensor is not elliptic. And that's the hyperbolicity of the whole system. So basically, if you have a basic wave equation, if you have the time, which is on the left side, and basically the space, which is on the right hand side, or whatever, this operator with a space derivative has to be elliptic for the system to be hyperbolic. And that is not true for the ADM equations. And so, okay, now that is... So basically, what then got me thinking is that that's a problem that sounded awfully familiar. Because earlier this century, actually in the early 20s, people had been playing with the four-dimensional equations basically when, you know, GR came hard from the press, okay, and they, you know, Lanchos and other people, and they, they don't know also, they realized that you can make a coordinate choice, namely harmonic coordinates, where, again, the exact same offending terms are identically zero, okay, and you can formally show that by again introducing what they call gauge source functions, okay, and then you can rewrite those offending terms as derivatives of the gauge source functions, okay, and then they realize you can set those gauge source functions to zero, which is harmonic coordinates, okay, and then in those harmonic coordinates the equations become manifestly hyperbolic. And so knowing that I realized that well, I got the same problem here so can you use the same solution? So I can here have the now it's a three-dimensional Ricci tensor which is not elliptic but I can make it elliptic by introducing these new functions. now they're not gauge source functions anymore as Helmut Friedrich pointed out to me because I don't use them to impose the gauge in my scheme I still use the labs and the shift to impose the gauge and I evolve those functions independently I have evolution equations for them however, so they're not zero and that's why they're not gauge source means that you use them to impose the gauge

7:30 and I'm not doing that, so that's the wrong technical term. But nevertheless, the way you write the Ricci chances in terms of those is exactly the same. But the key point is that you start out with having offending terms, which is the second derivatives of the metric. Now, introducing these functions, or these functions are first derivatives of the metric. and using those you can eliminate all the offending second derivatives of the metric and replace them by first derivatives of these auxiliary variables that is nice because that means that now the principal part of the operator is manifestly elliptic operator on the metric except you have them coupled to these auxiliary variables but only in lower order derivatives so it doesn't principal part of the operator anymore. You have to decouple it, basically. And then you evolve those other parts, and that's fine, but it doesn't spoil the ellipticity of that operator. And now, basically, I have never checked if the whole system is hyperbolic, so I cannot claim that, okay? But certainly, just staring at the operators, it just looks a lot nicer. You have an elliptic operator. It really is like a wave equation coupled But only first-order derivatives, okay? And so that's why that seems like something very natural to try. And then I did, and it worked just fine. And basically, when Scott came, we also actually coded up these herbs that Masaru, Shibata, Nakamura, those people use. And we actually get very similar results, okay? So basically they did something very similar, except that I believe, or personally I prefer my system, because the equations look a lot nicer than the operators are much simpler. It's just a lot simpler, if you compare it with it, it's a lot simpler. And also, you know, it's nice to have this historical background, you already know what you're doing. it's not being at hoc no that's a very interesting aspect and was it so was it that you kind of

10:00 spotted at some point as you were going along the connection to the Dondrich age and so on or did it happen that you've been recently looking back at these papers you know I don't even remember that anymore I was curious Okay, let me think about that for a second. Okay, I'm not sure in which order this happened, okay? But it is true that at the same time I started working on lecture notes, okay? And Stuart asked me to prepare, or to give a lecture series basically in his class on numerical relativity. So I put a lot of effort, actually a lot of work, into writing up lecture notes. And I did this, basically I used old notes. Andrew Abrahams did that many years back when I took the class and so I had his old notes and some other notes and stuff. so I base it in that but I add a lot of material by myself because I thought it's important and I actually wrote it up it looks nice it's typed up, it has exercises figures and what not ultimately we're hoping it's going to be a book actually anyways so at the same time this is roughly a year ago I prepared those lecture notes one thing that I put into lecture notes are harmonic coordinates and so I went through those arguments for the lecture notes and I realized that you can do that and then I made the connection but I don't even remember in which order that was but also oh no here's something else and that is Oh, no, yeah. Actually, many years back, there was also a meeting in Austin. Richard Matson invited Hamlet Friedrich to come and give a talk. I'm not sure if you were at the meeting or not. Probably not. And anyways, he did a very nice job. Basically, the whole thing was a little bit awkward, because here Richard Matson went ahead and invited Hamlet

12:30 Fridget to come and had him fly in from Berlin and so forth, and Howard Fridget didn't really know what this was all about and why they were interested in having him there. And Jimmy York felt offended because he kind of stepped on his toes because, you know, why does Richard Martzner invite this other mathematician? Doesn't he trust me? That kind of thing. I believe. Anyways, so what Helmut did is he gave a talk, a very basic talk, and how he explained how equations can be made hyperbolic. And the first thing that he talked about were those basically arguments by did-on-learn and so forth with the game source functions. So that's what I remembered. I think that's what I picked up on. So no, that is correct. So that's how I remember that you can do that. Then, oh, then there was another occasion. That is that basically this whole, then there's a Fisher-Marston business, right? Fisher-Marston, basically, they built up on the same formalism, but then made a first-order hyperbolic system out of it. Basically, if you only introduce the harmonic coordinates, you still have, basically, wave equation kind of system, you have second derivatives, and they made a first-order hyperbolic system out of that. and I had looked at that because at some point we were interested in maybe writing called Bass and the Fisher Marsden stuff actually that's right, that's what I first looked at, that's what I was first interested in and then I remembered talking to Mazzara and then I remembered what Talmud Fritje had talked about and then somehow it just clicked and I realized oh I can do the same thing in 3D So I guess I'm curious as to, with the principle reasons why putting equations in elliptic form is so convenient in the numerical case, as opposed to why... Okay, it's only the space part of the cylinder. Yeah, yeah, yeah. Well, nobody really knows. Okay, but certainly, or I believe nobody really knows. Maybe that's not it, but basically what it boils down to is

15:00 if you have a hyperbolic system, then you know that you can solve the problem, find numerical schemes which can solve that and you can stably evolve it. And basically you know that because that's the first exercise, at least I always do in that, is integrate the scalar wave equation. So even for this code, the first thing I did is I tried all the updating schemes just with a scalar wave. That works perfectly fine with outgoing wave boundary conditions, and it just works beautifully, no problem at all. And then you do the same thing with the ADM equations. and things go kabeng, you wonder why, and then if you stare at it, it's, well, the, I mean, fine, there are other terms there and some factors, but you don't really think that they should change the evolution that dramatically, but then if you look at it, the final difference operators, the difference operators, or differential operators are different, you simply, it's not, you know, it's certainly not, or it's not a hyperbolic system, it may need no one