Worskhop on initial value for binary black holes

Carlos Lousto, Albert Einstein Institute, Golm, Germany

lousto@aei-potsdam.mpg.de

lousto@aei-potsdam.mpg.de

The first (initial) workshop on initial data for binary black holes took place at the new location of the AEI: Golm, June 7-9 1999. It was extremely successful regarding the very high level of attendance and participation. Copies of the related papers (when available) prior to the meeting have been distributed as ``precedings'', and a certain flexibility in the schedule provided a fertile atmosphere for numerous questions and discussion.

The first day emphasized studies within the conformally flat ansatz for the 3-geometry. G. Cook presented a comprehensive review of the classical results and introduced new developments like the ``thin sandwich approach'' recently proposed by York[1]. He noted how this approach turns out to naturally include the approximations used by Mathews and Wilson and the ``convective'' one used by Lousto and Price in the particle limit. Cook also emphasized the need to move beyond the conformally flat ansatz and Bowen-York solutions of the momentum constraints in the quest to find more astrophysically realistic initial data. He also expressed interest in using the ``lambda'' systems described by Brodbeck, Frittelli, Hübner, and Reula for enforcing the constraints during numerical evolutions.

B. Brügmann presented the black hole punctures approach to initial data
based on Ref. [2]. Here the wormhole topology of black hole data is
compactified to *R*^{3}, which leads to an existence and uniqueness proof
and facilitates numerical studies.

C. Lousto described work with R. Price to test the Bowen-York initial data in the black hole plus a particle system, which can be treated perturbatively. The conclusion is that the ``longitudinal'' ansatz for the extrinsic curvature seems not to be a good representation of an astrophysically realistic scenario [3]. It was noted the interest on extending these studies from headon to generic orbits.

R. Beig described a formulation of the momentum constraints where quantities such as ADM linear and angular momentum are encoded by specific source terms in these equations concentrated at the punctures arising from conformally compactifying spatial infinity. He showed a way of writing down, in the conformally flat case, solutions for these inhomogeneous equations which are of the "Bowen-York"- type. One can then find the general solution, since the methods of Ref. [4] enable one to write down the general "unpunctured" TT-tensor.

The second day of the workshop started with an excellent review of the close limit approach to the final merger stage of two black holes by J. Pullin. He also discussed the last results on perturbations of black holes with angular momentum [5].

E. Seidel gave a comprehensive overview of the projects carried out in the numerical group at AEI. He described the results obtained by evolving (with the CACTUS code) black hole plus brill wave and pure brill wave data. For this latter data it is possible to follow either the scattering of waves leaving back flat spacetime or their collapse to form a black hole and then to obtain the corresponding quasinormal ringing.

M. Campanelli introduced the ``Lazarus project'' that proposes to marry full numerical techniques to describe well detached black holes with perturbative techniques to continue the evolution once a common horizon encompasses the binary system. The matching is performed by constructing the Weyl scalars on the perturbative slice [6], and evolve them using the Teukolsky equation.

C. Lousto then described the evolution of ``exact'' Misner data for two initially at rest black holes via the Zerilli and the Teukolsky equations. The results show that while the evolution with the Zerilli equation suffers a premature break down, the evolution with the Teukolsky equation is robust and generates results close to the linear initial data [7]. It Remains open the question whether this behavior also holds for non headon collisions.

R. Price neatly described his solution to the constraints representing two Kerr black holes in an axially symmetric configuration. He used the black hole plus brill wave form of the metric and superposed exact ``Kerr'' solutions to the momentum constraint. When one imposes this family of solutions, parametrized by the separations of the holes, to have a close Kerr limit, a ``pin pole'' between the holes appears [8]. Whether this is a generic feature or a consequence of the restrictive ansatz remains an open question.

W. Krivan showed the results of the evolution of the above initial data in the close limit regime where the ``pin pole'' is safely enclosed by the common horizon [9].

J. Baker discussed a similar approach to the problem, but he assumes the superposition of the holes at the level of the 3-metric and then solves for the extrinsic curvature [10]. Again, the requirement of the holes being kerr-like both when well separated and when close together generates unpleasant features such as discontinuities in the extrinsic curvature.

N. Bishop carefully described the Kerr-Schild ansatz to the initial value problem [11]. This is a quite original approach and has interesting potentialities. One needs still to identify the physical parameters and give explicit solutions for rotating black holes. He also mentioned the related Matzner et al work where two Kerr black holes are explicitly superimposed.

J. Thornburg [12] presented numerical initial data on an Eddington-Finkelstein slice of the spacetime with nonvanishing K. He stressed the benefits of working with 4th order evolution codes.

H. Shinkai exposed his work on how the post-Newtonian expansion can be used to give initial data for a further general relativistic evolution [13]. He applied this approach to binary neutron stars and measured the precision of several PN orders using as a criteria the violation of the Hamiltonian constraint. For the first and second PN expansion he gets around 60% and 45% errors respectively.

G. Nagy discussed about the differentiability of the Cauchy data
by studying the constraints in the case of incompressible
models of neutron stars. He proved rigorously its *C*^{1} behavior at the
boundary. Extension to more realistic equations of state is being undertaken.

P. Hübner presented a review of the conformal approach to GR and highlighted the stable numerical implementation of his evolution code (4th order) due to the explicit first order symmetric hyperbolic formulation [14]. Work is in progress on the question of giving astrophysical initial data and including matter sources.

On Thursday 10th we had two round tables on further work on initial data and other works on subjects related to gravitational radiation. This was planned to finish by 1 p.m. but given the interest of the participants it was extended to the whole afternoon with discussion of new ideas and even some computations!

Details and the full workshop program can be found in

http://www.aei-potsdam.mpg.de/~lousto/WID99.html.

**References**

[1] J.W. York, ``New data for the initial value problem of general relativity," gr-qc/9810051.

[2] S. Brandt and B. Brügmann,
``A simple construction of initial data for multiple black holes,"
Phys. Rev. Lett. **78**, 3606 (1997)

[3] C.O. Lousto and R.H. Price,
``Improved initial data for black hole collisions,"
Phys. Rev. **D57**, 1073 (1998)

[4] R. Beig,"TT-tensors and conformally flat structures on 3-manifolds",
in: Mathematics of Gravitation, Part 1, Lorentzian Geometry and Einstein
Equations (P.T. Chrusciel, Ed.), Banach Center Publications **41**, 109
(1997), also:
gr-qc 9606055.

[5] G. Khanna *et al.*,
``Inspiralling black holes: The Close limit,"
gr-qc/9905081.

[6] M. Campanelli, C.O. Lousto, J. Baker, G. Khanna and J. Pullin,
``The Imposition of Cauchy data to the Teukolsky equation. 3.
The Rotating case,"
Phys. Rev. **D58**, 084019 (1998)

[7] C.O. Lousto ``Linear evolution of nonlinear initial data for binary black holes: Zerilli vs. Teukolsky'', AEI-1999-7.

[8] W. Krivan and R.H. Price,
``Initial data for superposed rotating black holes,"
Phys. Rev. **D58**, 104003 (1998).

[9] W. Krivan and R.H. Price,
``Formation of a rotating hole from a close limit headon collision,"
Phys. Rev. Lett. **82**, 1358 (1999)

[10] J. Baker and R.S. Puzio,
``A New method for solving the initial value problem with application to
multiple black holes," Phys. Rev. **D59**, 044030 (1999)

[11] N.T. Bishop, R. Isaacson, M. Maharaj and J. Winicour,
``Black hole data via a Kerr-Schild approach,"
Phys. Rev. **D57**, 6113 (1998)

[12] J. Thornburg,
``Initial data for dynamic black hole space-times in (3+1) numerical
relativity," Phys. Rev. **D59**, 104007 (1999).
gr-qc/9801087.

[13] H. Shinkai, ``Truncated postNewtonian neutron star model," gr-qc/9807008.

[14] P. Hübner, ``A Scheme to numerically evolve data for the conformal Einstein equation," gr-qc/9903088.