The Lazarus Project:

Numerical relativity meets perturbation theory

Richard Price, University of Utah

Numerical relativity and gravitational wave detection exist in an entangled state. Though there are many opinions about this state, two things are probably not controversial. First, numerical relativity is required to compute the dynamics and gravitational radiation when inspiralling black holes merge, and second, that this is extraordinarily difficult. The difficulties prevent us from getting answers to questions that are not only crucial to determining the detectability of black hole mergers, but that are just plain interesting. Among these is the question of what happens when the binary pair, late in its inspiral, has too much total (spin plus orbital) angular momentum to form a Kerr hole. Does the inspiral stall? For inspiralling holes, is there a plunge or a gradual transition from slow inspiral to distorted final hole? Does the answer to this depend on such details as the spins? In the past year or two there has been a slow but steady advance of the frontier of the numerical relativity of binary black holes. True 3 dimensional runs have been successfully carried out for so-called grazing collisions [1,2] that describe non-axisymmetric collisions of two holes that start fairly close together with a fairly small impact parameter. Work is progressing in many centers of numerical relativity on a better understanding of the the outer boundary condition, how to excise the black hole (or avoid the need for excision), how to choose the numerical variables for greatest stability and/or accuracy, and much much more. This inspires confidence that it will not be too long before numerical relativity will be giving answers to astrophysical questions.

That confidence has taken a nice jump in the past few months. A group at the Albert Einstein-Max Planck Institute (``AEI'') has taken an eclectic approach, called the Lazarus Project [3] to looking at black hole mergers, and has provided waveforms generated by motion and merger after the holes move inward from the ``ISCO'' (the Innermost Stable Circular Orbit). The underlying idea is simple: If straightforward numerical relativity is used, the code evolving the spacetime would become unstable before useful information could be extracted. The Lazarus group therefore only uses numerical relativity where it is indispensable: to evolve the gravitational field from the ISCO stage to the point at which an almost stationary Kerr horizon is formed. After this, relatively simple black hole perturbation theory is used to continue the evolution of the spacetime. Perturbation theory allows evolution to rise from its unstable grave and to live again, like the biblical Lazarus. Sort of.

The initial Lazarus idea was the work of Carlos Lousto, Manuela Campanelli, and John Baker, who were joined early in the project by Bernd Brügmann. A student, Ryoji Takahashi, has also recently been added to the team. But the ``team'' in fact is the whole numerical relativity group at the AEI, since the numerical relativity code of the AEI and the CACTUS numerical relativity toolkit [4] constitute the front end of the eclectic Lazarus approach. The back end is the code to evolve perturbations of Kerr holes [5] developed several years ago. Not only did those elements already exist, but the idea itself of doing late stage evolution with perturbation theory is not new. The basic scheme had been set down in the mid 90s [6] and applied to simple axisymmetric processes [7]. What the Lazarus group did was to provide the tools for matching in a full 3D problem in which the final hole is a rapidly rotating Kerr hole. The details and difficulty of that task were what made this ``obvious'' step a real achievement.

The essence of the problem is to assign spacetime coordinates that are in some sense almost Boyer-Lindquist coordinates for the numerically evolved spacetime that is almost the Kerr spacetime outside the horizon, and in those coordinates to identify perturbations. There is, of course, no unique way of assigning coordinates and extracting perturbations. Small changes in the coordinates induce small changes in the perturbations. But the Teukolsky function, the quantity used in Lazarus perturbative evolution, is changed only to second order when small coordinate changes are made so the extraction process is insensitive if the spacetime geometry is really ``almost Kerr.'' The best way of building confidence about the method is inherent in the method. The matching of numerical relativity and perturbation theory is done at some transition time (i.e., on some spacelike hypersurface). If that transition time is taken to be too early, the numerically evolved spacetime will not have achieved the point of being a perturbation of Kerr. If that transition time is too late the numerical code will become unstable and the perturbations that are extracted will be unrelated to the physical problem. If there is a reasonable range of transition times that are neither too early nor too late, then the Lazarus method should give the same results for all transitions within this range. If the results are the same for a wide range of transition times, then it is very hard to resist accepting the results.

During development, the Lazarus method passed many tests. Among them was a test that results varied little for a range of transition times. But these were results for small transverse momentum, and hence were grazing collisions. The big question is whether numerical relativity plus Lazarus is ready to tell us about mergers, and the answer is either ``yes'' or ``very close.'' According to recent estimates [8,9], for puncture type initial data [10], the ISCO for equal mass, nonspinning holes, corresponds to a separation L of 4.8M (where M is the total ADM mass), a transverse momentum for each hole of 0.335M, and total angular momentum of $0.76M^2$. These initial data were used as the starting point for numerical evolution with an AEI code that uses maximal slicing, zero shift, and the standard ADM approach. The range of acceptable transition times was narrow, from about 10M to 12M, and the radiated energy generated was uncertain by a factor of around 2. It would be nice, of course, to hear of 10% accuracy, but a factor of 2 uncertainty is really quite respectable. Even with the limited accuracy the results carry some important astrophysical information. For one thing, there is an estimate of radiated energy 4-5%$Mc^2$, more than twice as large as any previous computation (though smaller than previous speculations), and a good omen for the detectibility of black hole mergers, and for the yet larger numbers that mergers of spinning holes may produce.) Aside from a welcome number, the results have an interesting qualitative lesson. There appears to be little radiation associated with the early motion of the holes; almost all the radiation can be ascribed to the dynamics of the distorted black hole that is formed.

It is possible, of course, that initial conditions that truly represent the ISCO will tell a rather different story (such as more orbital motion before the formation of a distorted horizon). To explore such questions what is needed is the ability to start evolution at an earlier time (or larger transverse momentum). This cannot be done at present; the numerical code used for evolution would go unstable before a perturbed Kerr hole is formed. But this will change, and that is perhaps the most exciting thing about the the addition of Lazarus to the set of numerical relativity toos. With Lazarus, the improvements in code stability that will be achieved in the coming months can quickly be turned into improvements in our understanding of black hole mergers.


[1] S. Brandt et al., Phys.Rev.Lett. 85 5496 (2000). [2] M. Alcubierre et al., preprint gr-qc/0012079.

[3] J. Baker, B. Brügmann, M. Campanelli, C. O. Lousto, and R. Takahashi, gr-qc/0102037; J. Baker, B. Bruügmann, M. Campanelli, and C. O. Lousto, Class. Quant. Grav. 17, L149 (2000).


[5] W. Krivan, P. Laguna, P. Papadopoulos, and N. Andersson, Phys. Rev.  D56m 3395 (1997).

[6] A. Abrahams and R. H. Price Phys. Rev. D 53, 1963 (1996).

[7] A. M. Abrahams, S. L. Shapiro and S. A. Teukolsky, Phys.Rev. D51, 4295(1995)

[8] G. B. Cook, Phys. Rev.  D 50, 5025 (1994).

[9] T. W. Baumgarte, Phys. Rev.  D62, 024018 (2000).

[10] S. Brandt and B. Brügmann, Phys. Rev. Lett. 78, 3606 (1997).

Jorge Pullin