I describe the construction of initial data for the Einstein vacuum equations
that can represent a collision of two black holes. I stress in the main
physical ideas.Comment: 5 pages, 2 figures. To appear in the Proceedings of the Spanish
  Relativity Meeting Gravitation and Cosmology ERE - 2002; isbn: 978844752738

Dain, S.

English

arXiv

I describe the construction of initial data for the Einstein vacuum equations that can represent a collision of two black holes. I stress in the main physical ideas

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Initial data for black hole collisions

Dain, Sergio

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arXiv:gr-qc/0212039 v1   9 Dec 2002Initial data for black hole collisionsSergio Dain11Max-Planck-Institut fu¨r Gravitationsphysik, Albert-Einstein-InstitutI describe the construction of initial data for the Einstein vacuum equationsthat can represent a collision of two black holes. I stress in the main physicalideas.The physical system we want to describe is a binary system of two black holes.It is expected that lot of such systems exist in the universe. Moreover, these systemsare expected to have the following two properties. First, gravitational radiation willbe emitted by them and the new detectors will be able to measure it. Second, thesesystems are completed described by the Einstein field equations. By this I mean thatthere exist solutions of the Einstein equations that can reproduce the gravitationalwave forms emitted by them. The ultimate goal is to find these solutions and comparethe waves forms with the result of the measurements.One method for constructing an appropriate class of solutions of the Einsteinequation is through an initial value formulation: we give appropriate data and thenwe evolved them numerically. It is not obvious that an initial value formulation is themost appropriate way to construct a solution for an astrophysical problem becausewe can not prepare the initial conditions of an astrophysical system in the laboratory.It is, in principle, impossible to find the exact initial data for a real astrophysicalsystem. The whole program will succeed if we are able to find some properties of thebinary system that do not depend very much on fine details of the initial data. Inthis sense black holes are perhaps better candidates than ordinary stars, since theyare much simpler, they do not involve the matter equations. For example, stationaryblack holes are expected to be characterized completed only by two parameters:mass and spin; on the other hand stationary stars can have a complicated multipolarstructure.A black hole is a region of no escape which does not extend out to infinity. In thestandard text books [6] [9] the definition of a black hole is made using the conformalcompactification of the space time. If the space time admit a suitable conformalboundary at null infinity and the causal past of this boundary is globally hyperbolic,then the black hole region is defined as the difference between the space time and thecausal past of null infinity. The boundary of the black hole region is a null surfacecalled the event horizon.I want to discuss the problem of finding initial data for black holes. An initialdata set for the Einstein vacuum equations is given by a triple (S, hab, Kab) where S12 S. DainS1S2Event HorizonBlak HoleregionApparentHorizonMatterCauhySurfaesGravitationalradiationFigure 1: Penrose diagram of a space time with a black hole.is a connected 3-dimensional manifold, hab a (positive definite) Riemannian metric,and Kab a symmetric tensor field on S. They satisfy the vacuum constraint equationsDbKab −DaK = 0, (1)R + K2 −KabKab = 0, (2)on S, where Da is the covariant derivative with respect to hab, R is the trace of thecorresponding Ricci tensor, and K = habKab. Examples of initial data are S1 andS2 in Fig. 1.We want to construct initial data set that evolve in a space time with a blackhole region. In order to do this, we must recognize the black hole on the initial data.The event horizon is not very useful because is a global property of the space time,we need to known the whole space time in order to calculate it. However, there isa remarkable consequence of the black hole theory, it is related with the concept ofthe apparent horizon. An apparent horizon can be defined as follows. Given a sliceS, we say that a two surface Σ contained in S is an apparent horizon if it satisfiesthe following equationDana−K + Kabnanb = 0, (3)where na is the unit normal vector to Σ. Equation (3) involves only quantities thatare present on the initial data. No evolution is needed. By the singularity theoremsInitial data for black hole collisions 3(see [6]) we know that a data with an apparent horizon will evolve in a geodesicallyincomplete space time. That is, the space time will have a singularity. If the cosmiccensorship conjecture is true, this singularity will be inside the black hole region.If there is a black hole, then the apparent horizon must be inside the black region.That is, the presence of an apparent horizon indicate the presence of a black hole.In general, a space time with a black hole will have slices with an apparenthorizon and slices without it. In Fig. 1 the slice S1 have no apparent horizon, andthe slice S2 contains one. Both data describe the same space time, but looking S1 itwill be very difficult to decide that the space time contain a black hole. The presenceof apparent horizon gives as a practical criteria to distinguish a black hole data.However, it is important to recall that in principle it is possible to have a black holewithout any apparent horizon.A space time contains two black black hole if the event horizon has two discon-nected components. Again, is very difficult to recognize this situation on the initialdata. But, we can recognize when the apparent horizon has two disconnected com-ponents. It seems to be reasonable to assume that when these two components ofthe apparent horizon are very separated from each other the event horizon will havealso two disconnected components.Summarizing: a simply way of constructing a two black holes initial data is byimposing that the data contain an apparent horizon with two disconnected compo-nents.The question is now how to construct initial data with apparent horizons. Inorder to force the data to have an apparent horizon we have to impose to the con-straint equations (1), (2) appropriate boundary conditions. Black hole data will bejust a particular choice of boundary conditions.SAsymptotiFlatnessInnerBoundary12Figure 2: A black hole initial data with an apparent horizon with two disconnectedcomponents.The boundary condition are divided naturally in two: an outer boundary con-dition at infinity and an inner boundary condition. The outer boundary condition isasymptotically flatness, it means that we are dealing with an isolated system. It is afall off condition for the metric hab and the extrinsic curvature Kab. This boundarycondition is well understood. The inner boundary condition is the condition thatwe have to impose on the inner boundary in order to force it to be an apparent4 S. Dainhorizon. The inner boundary condition is not so well understood as the outer one.The standard way to produce an apparent horizon involves the choice of a non trivialtopology for the data [2] [7] [1]. These methods introduce the apparent horizon in anindirect manner. They do not really deal with an inner boundary. I think it wouldbe very interesting to know how to prescribe directly this inner boundary conditionin a consistent way. The only article I known in this direction is a numerical studyby J. Thornburg [8].There exist an infinite number of data which contain apparent horizon with twodisconnected components. Which is, among them, the data that will correspond tothe real astrophysical process? If the black hole are close to each other to answerthis question is probably as difficult as to calculate the whole space time. But if theblack hole are initially very well separated from each other then one expect somesimplifications. In particular, the first requirement we should impose is that the datanear each of the black hole should be, in some appropriate sense, close to the data ofone isolated black hole. We call this the far limit of the data. The important point isthat we know exactly the data for one black hole: the Kerr initial data. This suggestthat one way to produce an initial data with the right far limit is by “summing” twoKerr initial data.If the black holes do not have spins, then they should have a far limit to theSchwarzschild one. The Schwarzschild initial data can be chosen to be time sym-metric, that is Kab = 0. With this assumption the constraint equations can bereduced to a linear equation. In this case is simpler to make sense of “summing” twoSchwarzschild black holes, this has been done in [2] [7].In the case of Kerr, the constraint equation are non linear. It is possible tosum part of the initial data (the “free data”), for the rest of the data one shouldsolve the constraint equations. One can prove that, under certain assumptions, theseequations have solutions. Then, one obtain an initial data which have a far limit tothe Kerr initial data. This construction has been made in [5] [4]It important to note that in all these data the linear momentum of the individualblack holes is zero in the far limit approximation. There exist data with arbitrarylinear momentum, like [1], but they do not have a fat limit to Kerr. They are alsonumerical studies of data which have arbitrary linear momentum and also a far limitto Kerr (see the review [3] and reference therein), but so far no analytic existence proofof these type of data is available. The main difficulty is that most of the knowledgewe have about the constraint equations is based on the assumption that the datais maximal (i.e; K = 0). The Kerr data in standard Boyer-Lindquist coordinates ismaximal. However, remarkably enough, is it not known any maximal, boosted (i.e;with no trivial linear momentum), initial data for Kerr or even for Schwarzschild.These boosted maximal slices can in principle be used to construct a data for twoblack holes in which each of them has arbitrary linear momentum in the far limit.Initial data for black hole collisions 5References[1] J. M. Bowen and J. W. York, Jr. Time-asymmetric initial data for black holesand black-hole collisions. Phys. Rev. D, 21(8):2047–2055, 1980.[2] D. Brill and R. W. Lindquist. Interaction energy in geometrostatics. Phys. Rev.,131:471–476, 1963.[3] G. B. Cook. Initial data for numerical Relativity. Living Rev. Relativity, 3(5),2001. http://www.livingreviews.org/Articles/Volume3/2000-5cook/.[4] S. Dain. Initial data for a head on collision of two Kerr-like black holes with closelimit. Phys. Rev. D, 64(15):124002, 2001. gr-qc/0103030.[5] S. Dain. Initial data for two Kerr-like black holes. Phys. Rev. Lett., 87(12):121102,2001. gr-qc/0012023.[6] S. W. Hawking and G. F. R. Ellis. The large scale structure of space-time. Cam-bridge University Press, Cambridge, 1973.[7] C. W. Misner. The method of images in geometrostatics. Ann. Phys. (N.Y.),24:102–117, 1963.[8] J. Thornburg. Coordinates and boundary conditions for the general relativisticinitial data problem. Classical and Quantum Gravity, 4(5):1119–1131, 1987.[9] R. M. Wald. General Relativity. The University of Chicago Press, Chicago, 1984.

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Initial data for black hole collisions

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