arXiv:gr-qc/0206003v1 2 Jun 2002 ∗ ilntrqiei ntefloig htmasta in that condition means the That horizon, following. the and the of practice, in point in it each verify require out- not to an will difficult be I is it surface that such condition ermost The expansion zero topology. with spherical slice) and (time hypersurface spacelike a paethrznfidrta scoeyitgae with integrated closely robust is code. re- and evolution that excised fast the a finder the have to horizon of important apparent shape therefore is and It used location be gions. to the has determine horizon apparent to numeri- the a where in evolution, conditions cal boundary excision apply to wants spin. pos- and then mass it their cir- calculate making to horizons, certain sible isolated under horizon. also apparent are cumstances the horizons horizon outside apparent event or Furthermore, an at is under somewhere there indicates, present that horizon a apparent assumptions, during each not reasonable is Yet them spacetime find the known. of to shock yet future possible the the because not evolution, as is time such it global and are time, horizons structures, Event in hydrodynamics. locally in found fronts determined that features be prominent other can no In has spacetimes. spacetime relativistic vacuum, general of evolution numerical saeie uwr omlt h oio,and horizon, the to normal outward (spacelike) a ohl seeg 1,catr4,o 1].Here [16]). or 4], chapter [15, e.g. (see hold to has K ieslice. time mi:[email protected] Email: ij naprn oio sa uems -ufc within 2-surface outermost an is horizon apparent An oaigaprn oiosi loncsaywe one when necessary also is horizons apparent Locating the during purposes several serve horizons Apparent r h he-ercadetisccraueo the of curvature extrinsic and three-metric the are H H oiosfrsnl n utpebakhl configurations. hole black multiple singul and coordinate single polar for the of detai horizons qua treatment intermediate in the the explain and of conditions components I tensor the methods. represent standard and horizo with apparent The equation implementation. elliptic efficient an to components leads h coordinate and the Cartesian describing using represented quantities reso are tensor horizon yet the arbitrary that is for approach allowing this surface, horizon the of steso-called the is rsn atagrtmt n paethrzn.Ti alg This horizons. apparent find to algorithm fast a present I := .INTRODUCTION I. ∇ i I METHOD II. s i + K ij hoeiceAtohsk Universit¨at T¨ubingen, 72076 Astrophysik, Theoretische ( paethrznfunction horizon apparent s i s emn.Wb http://www.tat.physik.uni-tuebingen.de Web: Germany. j − ataprn oio algorithm horizon apparent fast A g ij (1) 0 = ) s Dtd 2002-05-12) (Dated: g i ij rkSchnetter Erik sthe is and . a coordinates cal radius the choice) arbitrary hsrpeetto ed oteequation the to leads representation This h condition the hog nexpression an through lctl yafunction a by plicitely ie ag peea nta aa hi disadvantage Their data. initial as sphere large a given datgso o ehd r htte r ngen- are the in they find that are to and able horizon, they practice an that finding in in are The robust more methods [2]. eral flow procedures minimisation of or advantages level or [14], [13] Com- other methods flow curvature also literature. flow mean the are are in methods there found used do, be monly to I it as solving for equation, methods elliptic an as tion te hie r qal osbe h oio consists ( horizon points The those practice; of possible. in out equally problems are cause choices not origin, other does the restriction about regions this star-shaped but of those to surfaces tion o h uwr oml h function The normal. outward the for F fteequation the of fa loih,s htId att ra hmi some in them treat to want do I that detail. implementation so numerical algorithm, the an for of important rather ally ( ordinates ( coordinates normal like ihcodnt iglrte ttepls h ag of range the poles; is the equation at the singularities coordinate with h scnb enb usiuigtedfiiin of definitions the substituting by seen be can as , h paethrzncnas edfie implicitly defined be also can horizon apparent The naprn oio ufc a erpeetdex- represented be can surface horizon apparent An satraie otetn h paethrznequa- horizon apparent the treating to alternatives As h condition The into F ∗ uin n,i rnil,sae.Nvlin Novel shapes. principle, in and, lutions rznlv ietyo h oio surface, horizon the on directly live orizon hseiiae oriaesingularities, coordinate eliminates This . rte.Ls iea xmlsapparent examples as give I Last arities. H endeg as e.g. defined h oriaessesue ostore to used systems coordinate the l tte,addsrb h rdboundary grid the describe and ntities, qaini hnsle sanonlinear a as solved then is equation n ihteaoecodnt hie h domain the choice, coordinate above the With . rtmue nepii representation explicit an uses orithm ,y z y, x, s ,θ φ θ, r, i r F θ a ecluae sn ihrspherical either using calculated be can h = .Sc oriaedsicin r actu- are distinctions coordinate Such ). H H and ( ≡ s ,θ φ θ, r, h ,o rnfre oueCreinco- Cartesian use to transformed or ), ( i ( sannierelpi qainin equation elliptic nonlinear a is 0 = ,φ θ, T¨ubingen, r ,φ θ, = ,θ φ θ, r, φ ∈ F hscoc etit h possible the restricts choice This . s( is 0 = ) p = ) h ). (0 ( ( x g , ,φ θ, ftetm lc htsatisfy that slice time the of ) i jk ∞ ,wt h ee e func- set level the with 0, = ) r outermost r ∇ ∇ − ). htseie ti san is (this specifies that ) safnto ftespheri- the of function a as i j F h F ( ∇ ,φ θ, ,φ θ, k F ) (2) . ) paethorizon, apparent ∈ F (0 n h space- the and π , ) × [0 s i , and 2 (3) π ) 2 is that they lead to parabolic or degenerate elliptic equa- 4 tions that are expensive to solve. Minimisation proce- 3 dures can also lead to a more stable formulation than elliptic formulations, but are usually slower. I will re- strict myself to the solution of the elliptic equation. As 2 described below, I have also tried a simple Jacobi solver that is conceptually identical to a flow method, and it shows the expected robustness. It is possible to change the nature of equation (1) by extending it into the three-dimensional time slice. This is 1 e.g. done by changing from the unknown function h(θ, φ) 0 to a level set hλ(θ, φ), where λ is the level set parameter [13]. The advantage of such an extension is that multi- Figure 1: North pole of the grid on the horizon, in a polar projection. This shows the grid lines as they lie on the sphere. ple horizons can be found at the same time, and that no initial guess for the horizon location is needed. The dis- advantage is that such an extension will lead to a slower implementation because of the additional dimension. Let me use the indices i, j, k for Cartesian components (i, j, k [x,y,z]), u, v, w for spherical components in 3D (u,v,w∈ [r, θ, φ]), and a, b, c for spherical components on the horizon∈ (a,b,c [θ, φ]). This convention distin- III. COORDINATES guishes clearly which quantities∈ are actually defined on what manifold, and also makes coordinate transforma- The above choice to use spherical coordinates to pa- tions explicit. rameterise the horizon introduces a preferred coordinate system into the otherwise coordinate-independent equa- tions (1), (2), and (3). Another preferred coordinate sys- IV. DISCRETISATION tem comes from the grid used in the time evolution code for the whole spacetime. Often, gij and Kij are given in Cartesian components on a Cartesian grid. This raises The discretisation scheme of the 3D quantities, i.e. the question as to which coordinate system to use in the those living in the whole time slice, does not play a role numerical implementation of the horizon finder. The co- when locating a horizon. There only has to be a way ordinate choice will of course not influence the physical to interpolate these quantities onto the horizon surface. results, but it can make the implementation more com- These quantities are usually discretised on a Cartesian plicated if it requires interpolation, or less accurate if it grid. It is also possible to use spherical coordinates, or leads to coordinate singularities. It can also make the to use mesh refinement, without influencing the way in results more or less difficult to interpret, as tensor quan- which the apparent horizon function H is evaluated. tities are numerically always expressed through their co- I chose to discretise the 2D quantities, i.e. those quan- ordinate components. Tensors in different coordinate sys- tities living on the horizon, by using a polar θ-φ-grid with tems cannot easily be compared, even if they are given constant grid spacings dθ and dφ. A constant grid spac- at the same grid points. ing works well when the horizon has a shape that is not I chose to represent the quantities on the horizon us- too far from a sphere. Experiments show that distorted ing their Cartesian tensor components, even when they (peanut-shaped) horizons still work fine. are given on grid points on the horizon surface. This has Polar coordinates create coordinate singularities at the advantage that the tensors gij and Kij do not have θ = 0 and θ = π, which I avoid by having the grid to have their components transformed, although they do points staggered with respect to the poles. I find that need to be interpolated to the apparent horizon location. putting scalar quantities on such a grid generally works It also has a disadvantage, because the partial deriva- fine and requires no special treatment near the poles. tives that are calculated on the horizon are given only On the other hand, naively putting tensor components in spherical components, and those have then to be ex- in spherical coordinates on such a grid is a bad idea, as plicitely transformed into Cartesian components. they either become zero or diverge at the poles. The overall advantage of using Cartesian tensor com- The boundary condition in φ-direction is periodicity: ponents is that non-scalar quantities on the horizon can f(θ, φ) = f(θ, φ +2π) for any function f living on the more easily be compared to quantities given in the whole horizon. The boundary condition in θ-direction, i.e. spacetime, and that the coordinate singularities at the across the poles, is a bit more involved. It is f(θ, φ) = poles do not influence the representation of such quan- P f( θ, φ+π) for arbitrary tensor components f, where tities. The singularities will of course still influence the the· parity− P = ( 1)r depends on the rank r of the tensor calculation of quantities on the horizon, but the represen- of which f is a component.− Figures 1 and 2 demonstrate tation of the results will not have coordinate singularities the polar boundary condition, especially how the shift by any more. π in φ-direction comes about. 3

V. EVALUATING THE APPARENT HORIZON FUNCTION H

The apparent horizon finder uses the algorithm de- North pole scribed below to evaluate the apparent horizon func- tion H. This description lists the important interme- diate quantities used in the apparent horizon finder, and explains the order in which they are calculated. This method to evaluate H is then subsequently used to find 0 1 2 3 4 a solution for h. Although the quantities F , si, and H are defined ev- Figure 2: North pole of the grid on the horizon, as seen from erywhere in space, they only need to be evaluated on the the grid. One can see that the black and white dots have horizon, i.e. on the surface determined by h. They still multiple images on the coordinate grid, separated by π in the need to be defined in a neighbourhood of the horizon, φ-direction. so that their r-derivatives can be taken. Their continu- ation off the horizon is chosen arbitrarily, and is implic- itly determined by the above choice of spherical coordi- This grid allows numerical partial derivatives to be nates. Thus the finder calculates quantities on the two- taken only in the θ- and φ-directions. The apparent hori- dimensional horizon surface only, although the quantities zon equations (1), (2), and (3) from above also need r- live in three dimensions. derivatives, and those cannot be taken numerically. How- It would be a bad idea to calculate the derivative of si ever, the r-derivative of the coordinate transformation numerically, as si is itself already calculated as a deriva- operators are known analytically, and one can see from tive. Taking a derivative of a derivative leads to an effec- the definition of F that ∂rF = 1 by construction. Fi- tive stencil that is larger, and hence slower, and that con- nally, the partial derivative of the three-metric, which is tains elements with a weight of zero. These zero weights given in Cartesian coordinates, is calculated before the lead to an odd–even decoupling of the grid points, which metric is interpolated onto the horizon. in turn leads to large errors. Instead, one uses the chain i My choice of evaluating the apparent horizon function rule, and calculates is directly from second derivatives directly on the grid that forms the horizon surface seems of F . This makes it∇ necessary to calculate numerically i i to be an uncommon one. It is also possible to evaluate i j F and is along with iF and s . ∇ ∇ ∇ ∇ the apparent horizon function not on the surface itself, I describe explicitely which quantities are actually but instead in the three-dimensional spacetime on those stored, and in which coordinate system the tensor quan- grid points that are close to the horizon [11, 14, 16]. This tities are calculated. method requires interpolating in both directions between the surface and the time slice. It has the advantage that it is basically independent of the apparent horizon dis- Algorithm cretisation, but it is also more expensive. In addition, the domain of the equation, i.e. the set of active grid points, 1. Give, as fixed background quantities, the three- changes with the horizon surface, leading to further com- metric g (xk) and the extrinsic curvature K (xk) plications. ij ij in the whole time slice. These quantities do not While I represent the apparent horizon shape using an change while the horizon is located. explicit grid, one different commonly used way is to ex- pand the horizon surface location in spherical harmonics 2. Determine a 2-sphere h(xa) for which the apparent [1, 2, 8, 10]. Compared to using a grid on the hori- horizon function is to be evaluated. This is done by zon surface, an expansion in spherical harmonics has the an elliptic solver as described below, and is outside advantage of having no coordinate singularities. Fur- the scope of this algorithm. thermore, one has direct insight into and control over the high spatial frequency components. Such an expan- 3. Calculate the horizon grid point locations in Carte- sion would lend itself naturally to a multigrid algorithm. sian coordinates as xi(xa), using the definition of A multipole representation is by principle restricted to h. star-shaped regions, while an explicit representation is generic; however, this disadvantage is not important in 4. Interpolate the three-metric, the extrinsic curva- practice. More problematic is that integrations over the ture, and the partial derivative of the three-metric 4 surface are rather expensive, as they scale with O(lmax), onto the horizon. These quantities can conceptu- u u were lmax is the number of multipole moments used. ally be viewed as representing gij (x ) and Kij (x ), a a a With an explicit grid, these integrations scale with O(n), but are stored as gij (x ), Kij (x ), and ∂kgij (x ). where n is the number of grid points, and where one (The partial derivative is needed later.) Note that 2 should choose n = O(lmax) for a fair comparison. these tensors now live on the horizon, but have their 4

components still in Cartesian components, prevent- in practice and is called apparent horizon tracking. How- ing coordinate singularities near the poles. ever, finding horizons in an unknown spacetime is more difficult. Depending on how e.g. the initial configuration u 5. Define the implicit horizon location F (x ) = for an evolution run is constructed, one does not know F (r, θ, φ)= r h(θ, φ). This quantity is not stored − whether, where, or of what shape the horizons are. A explicitely. similar problem is encountered during the evolution, as u new apparent horizons may appear. One does not know 6. Calculate an outward-directed covector vF (x ) that is orthogonal to the surface. By definition,∇ their location and not even their time of appearance in advance. ∂rF = 1, and ∂aF = ∂ah. This is stored as a a − Shoemaker et al. [14] have implemented a mechanism ∂vF (x ) and ∂v∂wF (x ). (The second derivatives are needed later). The calculation needs the values in their apparent horizon finder that automatically de- a a tects whether a two black hole system has a common ∂bh(x ) and ∂b∂ch(x ), which are calculated on the horizon surface. horizon, and if not, finds the two separate horizons in- stead. I did not use such a mechanism; if a horizon does u 7. Define the transformation operator Ti that trans- not exist, I had my solver abort. I found it to be sufficient forms from spherical to Cartesian covariant coordi- in practice to set up the finder manually with different u nate components. This operator is given by Ti = initial guesses, corresponding to the different suspected ∂xu/∂xi, where the xi(xu) are given by the usual horizons in a time slice.

x = r sin(θ)cos(φ) y = r sin(θ) sin(φ) A. Newton-like method z = r cos(θ) I implemented a fast Newton-like method by calling the Evaluate this operator and its derivative at r = h, PETSc library [6, 7]. This library provides several effi- u a u a and store it as Ti (x ) and ∂j Ti (x ). cient parallel solvers for nonlinear elliptic equations. It uses Newton-like methods to reduce the nonlinear prob- 8. Transform the components of the outward-directed lem to a linear one, and then offers Krylov subspace u covector into spherical coordinates: iF (x ) = methods to solve it. v u a ∇ a T vF (x ). Store iF (x ) and i j F (x ). i ∇ ∇ ∇ ∇ Additionally, PETSc offers to calculate numerically the The calculation of the second derivative needs the Jacobian that is necessary for a Newton-like method. I derivative of the transformation operator T , and estimate that this is probably about an order of magni- also needs the partial derivative of the three-metric tude slower than a hand-coded Jacobian; it is therefore for the connection coefficients. not recommended by the PETSc authors. On the other hand is it much easier to implement, and most impor- 9. Normalise the outward-directed covector F (xu) i tantly also much easier to get correct. Thornburg [16] and raise its index to si(xu), using the∇ three- explains the necessary and tedious steps of implementing metric g . Calculate its derivative si(xu). This ij i an explicit Jacobian in detail. Huq et al. [11] use numer- needs the second derivative of F . Store∇ si(xa) and i a ical perturbations instead, and their explicit calculation is (x ). ∇ of those seems to complicate their implementation signif- 10. Calculate the apparent horizon function H = icantly. Fortunately, PETSc is fast enough in my case, i i j ij a is + Kij (s s g ). Store it as H(x ). so I never tried a hand-coded Jacobian. Tracking a hori- ∇ − zon with a reasonable resolution takes only about half a For the above calculation I use a second-order uniform second on a notebook with yesteryear’s hardware. Cartesian interpolator, and second-order centred differ- ences to approximate the partial derivatives. Altogether, the apparent horizon function H thus depends on the B. Flow-like method background spacetime as given by gij and Kij , and de- pends on the horizon location h. Unfortunately, the apparent horizon equation is rather nonlinear, and the radius of convergence of Newton- like methods seems to be very small. Instead of a fast VI. FINDING THE HORIZON Newton-like method, one can apply a slow but robust flow-like method, which can be viewed as essentially an In order to solve a nonlinear equation, one needs initial iterative Jacobi-type relaxation method. Such a method data, which is in this case an initial guess for the horizon has a larger radius of convergence, and is in practice reli- location. This initial data selects which apparent horizon able in finding the outermost apparent horizon. For this will be found, if there are several present in the time slice. method one generally uses a large sphere as initial data. During the evolution one can easily use the location from The horizon shape then flows gradually inwards, follow- the previous time slice as initial data; this works very well ing the expansion of the horizon surface [14]. 5

With the above coordinate choice, the grid spacing run dx dφ R/2 MADM near the poles is very small, and this requires very small a 1/4 2π/36 0.991111 1.00043 −3 convergence factors of the order of 10 or less for a rea- b 1/8 2π/72 0.997693 1.00010 sonable resolution. Finding a horizon with a coarse res- c 1/16 2π/144 0.999429 1.00002 olution takes several minutes on the above hardware. In 0 0 1.0 1.0 principle one could first flow-find a coarser approxima- tion to the horizon, and then Newton-find the horizon Table I: Horizon and ADM masses for a black hole with M = 1 with a higher resolution, but I have not tried this. and a = 0. Both the spacetime and the horizon surface vary in resolution. The last row gives the analytic values. I do not think that a multigrid algorithm can be used to accelerate flow-finding, because the coarse resolution is already so coarse that no coarser grid could reasonably runs R/2 MADM describe a spherical shape. It might be worthwhile to use a, b, c 3.79 4.43 a different discretisation scheme for the horizon surface, a, b 3.85 4.37 so that the grid spacing does not tend to zero near the b, c 4.04 4.18 poles, or to use spherical harmonics as mentioned above. Table II: Convergence factors for the results from table I. A value of 4 indicates perfect second order convergence.

VIII. EXAMPLES VII. ANALYSING THE HORIZON

I have implemented this apparent horizon finder al- Once the location of the horizon is known, one can gorithm within the Cactus framework [5], which has re- calculate its area A by integrating over the surface, and cently gained some popularity in the numerical relativity from that the horizon radius R via the definition A = community. This will potentially make it easier for other 2 4πR . The irreducible mass is given by R/2, which is people to use my implementation of this algorithm not different from the total mass M in case the spin a is only as an external programme, but also directly from non-zero. within their code. One can also calculate the lengths of the equatorial Below I give results from applying this finder to con- and two polar circumferences (at φ = 0 and at φ = π/2). figurations with one and multiple horizons. First I use These circumferences can give a hint as to the coordinate single black holes in Kerr–Schild coordinates as test data distortion of the horizon. Of course, these hints are only to judge the accuracy and quality of this method. Then reliable if one knows in advance about the symmetries of I apply the finder to superposed Kerr–Schild data to the horizon, which is only the case in testing setups. demonstrate the behaviour of this method in the pres- ence of multiple black holes, and compare the horizon If one assumes that the horizon has a spin that is masses to the corresponding ADM quantities. aligned with the z axis, then one can use the area and the equatorial circumference to calculate the spin mag- nitude. In the elliptic coordinates that one prefers for A. Single black hole Kerr black holes (see e.g. [9, section 3.3]), a horizon with mass M and spin a is located at the coordinate radius I performed a convergence test with Kerr–Schild data. r = M + √M 2 a2. It has the area A = 4π(r2 + a2) − 2 2 (See e.g. [9, section 3.3.1] for a definition of the Kerr– and the equatorial circumference L = 2π(r + a )/r. (I Schild coordinates.) Table I contains results from runs thank Badri Krishnan for pointing out the latter to me.) with a black hole of mass M = 1 and spin a = 0. The These equations can be solved for a and M. resolution of both the 3D time slice and the 2D grid on It turns out that the estimate for the total mass M that the horizon increases downwards. The last line contains is calculated this way has a reasonable accuracy, while the analytic values. R/2 is the irreducible horizon mass, the estimate for the spin a is rather inaccurate when a is and MADM is the ADM mass (see e.g. [17, eqn. (7.6.22)]) small. In this case, a2 can even become negative. This is of the whole spacetime, which I calculated numerically mainly caused by accumulation of numerical errors. An for reference. O(10−2) error in a2 shows naturally up as an O(10−1) In general, I would consider the resolutions 1/4, 1/8, error in a. Of course, the values for a and M are only and 1/16 to be coarse, reasonable, and fine, respectively. correct when the spin is indeed aligned with the z axis, Corresponding time evolution simulations need to run in and that will usually not be the case (nor will it be easily the year 2002 on a notebook, a workstation, or a super- verifiable) in a spacetime that is given numerically. How- computer. ever, if the horizon should be isolated, then the isolated Table II shows the convergence factors for three-way horizon formalism [3, 4] can provide a way of reliably cal- and two-way convergence tests from the above runs. The culating the spin and thus the total mass of the horizon. convergence factors f are calculated in the usual way via 6

run dφ R/2 az M plain A 2π/36 0.960886 0.496507 0.994655 2 az=1/2 a =1/2 v =1/2 B 2π/72 0.964675 0.498183 0.998537 z x az=1/2 vz=1/2 C 2π/144 0.965594 0.498815 0.999511 singularity D 2π/288 0.965819 0.498997 0.999754 1 0 0.965926 0.5 1.0

Table III: Horizon masses and spins and ADM masses for a z 0 black hole with M = 1 and az = 1/2. Only the horizon sur- face resolution varies. The last row gives the analytic values. -1

runs R/2 az M A, B, C 4.13 2.65 3.99 -2 B, C, D 4.08 3.47 4.01 -2 -1 0 1 2 3 Table IV: Convergence factors for the results from table III. x A value of 4 indicates perfect second order convergence. Figure 3: Apparent horizons for Kerr–Schild black holes with varying spins and boosts. Displayed is a cut through the x-z f = (C M)/(M F ) or f = (C A)/(F A), where plane. C, M, F−, and A represent− the coarse,− medium,− fine grid, and analytic values, respectively. A factor of 4 indicates perfect second-order convergence. The results indicate straint equations. This results in spacetimes with either that a spatial resolution of dx = 1/4 is not sufficient to two separate or a single distorted (“merged”) black holes. be in the convergence regime, and also that the numerical One can freely specify the locations, masses, and spins of ADM masses reported here might not be very accurate. the black holes prior to the superposition. The relation In another convergence test, shown in table III, I to the properties of the superposed black holes is not kept the spatial resolution of the time slice constant at known, but it is hoped the superposition will not change dx =1/8, and varied only the resolution of the apparent them much. horizon grid. This time the black hole had a mass M =1 For the superposition, I chose a grid spacing of dx = and a spin az = 1/2, again in Kerr–Schild coordinates. 1/4, an outer boundary that has a distance 8.0 from the The apparent horizon spin az is estimated via its equa- centres of the black holes, and excised a spherical region torial circumference. The last line of the table shows the with radius 1.0 around the singularities. I superposed the analytic values. three-metric and the extrinsic curvature without atten- The convergence factors for this test are shown in table uation. I then solved the constraint equations using the IV. As the spacetime is now fixed, and differs slightly York-Lichnerowicz method with a conformal transverse- from the analytic solution due to the discretisation errors, traceless decomposition. (This method is described e.g. I omit the two-way convergence tests. It is clearly visible in [9], section 2.2.1.) I used Dirichlet boundary condi- that the spin estimate does not converge to second order. tions for the vector potential and the conformal factor, This is unfortunate, but not of much consequence, as keeping the boundary values from the superposition. this method for the spin calculation is not applicable in Whether or not a common apparent horizon exists is a general anyway. hint as to whether the black hole configuration contains Figure 3 shows horizon shapes as reported by the results from two separate or a single merged black holes; finder. It shows apparent horizons of four black holes this fact is not known otherwise. The only way to find out with varying spins and boosts. The singularity is at the would be to evolve this configuration for a long enough origin for a = 0, and is a ring with radius az in the time so that the event horizon could be tracked back- x-y plane when the black hole is spinning. This figure wards in time. This is unfortunately not easily possible shows nicely that excision boundary conditions become (if at all) with today’s black hole evolution methods. more complicated for spinning black holes, as the appar- I created a series of initial data configurations for two ent horizon and the singularity can get rather close to equal-mass (M = 1) non-spinning non-boosted black each other. holes with varying distances. Table V shows the com- mon horizons masses MC = RC /2, the individual hori- zons masses MI = RI /2, and the ADM masses of the B. Multiple black holes spacetimes MADM. The spins of the superposed black holes are known to be zero for symmetry reasons. In order to prepare and analyse initial data for binary The row z = 0.0 contains two black holes that were black hole collision runs, I superposed two Kerr–Schild superposed at the same location, i.e. form a single black black holes as proposed in [12], and then solved the con- hole in a coordinate system different from Kerr–Schild. 7

z RC /2 RI /2 MADM EB ER 4 0.0 2.280 — 2.306 — +0.026 inner horizons common horizon 1.0 2.245 ? 1.973 ? -0.272 3

1.75 2.167 1.137 1.986 -0.107 -0.181 2 2.0 2.130 1.087 1.989 -0.044 -0.141 1 2.5 2.023 1.041 1.994 -0.059 -0.029

3.0 (2.101) 1.020 1.997 (+0.060) (-0.104) x 0 4.0 — 1.002 2.000 -0.004 -1 6.0 — 0.991 2.000 +0.018 8.0 — 0.989 2.000 +0.022 -2 -3 Table V: Masses of the common horizons, the individual hori- zons, and the ADM masses for two superposed black holes -4 for varying distances. EB is binding energy of the two black -6 -4 -2 0 2 4 6 holes, and ER is amount of radiation present in the spacetime, z as explained in the text. For z = ±3.0, no common horizon was detected any more. The last three rows show the sum of Figure 4: Common and individual horizons for superposed ± EB and ER only. Kerr–Schild data at z = 2.0.

4 This is a consequence of the way the extrinsic curvatures inner horizons common horizon are superposed. Note that the mass is not simply twice 3 the mass of a single black hole. Also, as this spacetime is not spherically symmetric due to the outer boundary 2 condition, the ADM mass and the horizon mass need not 1 be the same. In the row z = 3.0, the apparent horizon finder did x 0 ± not converge, but was close. The L2-norm of the residual -1 was still 0.0248 when the solver aborted. That means that from this row on there is no common horizon any -2 more, but the solution error here is still comparable to -3 the discretisation error. -4 The difference EB = MC 2MI can be interpreted as − -6 -4 -2 0 2 4 6 the binding energy of the system. (Here MC = RC /2 z and MI = RI /2 because the spin is zero.) The binding energy magnitude seems to decrease with distance, and Figure 5: Common and individual horizons for superposed is negative as it should be, except in the row z = 3.0 Kerr–Schild data at z = ±1.75. The individual horizons do where there was no common horizon detected any more.± overlap, and are surprisingly only minimally distorted. The difference ER = MADM MC is the amount of − radiation present in the spacetime. Of course, ER must be positive, meaning that the negative values found nu- Figure 4 shows the apparent horizons of two super- merically are unphysical and point to numerical errors. I posed black holes with centres at z = 2.0. The common suspect that the error lies mainly with the calculation of apparent horizon indicates that there± is a common event the ADM mass. I calculate the ADM mass as a surface horizon present, and that this time slice thus contains a integral near the outer boundary. When I evaluate the in- single, merged black hole. In figure 5, the black holes are tegral at locations further inwards, closer to the apparent closer together at z = 1.75. The two inner apparent horizon, then the ADM mass increases and gets closer to horizons actually do overlap;± this is not a numerical er- the horizon mass. This is inconsistent and cannot happen ror. It is surprising to see that the shape of the horizons when the constraints are satisfied. However, in a numer- is not distorted more. But unlike event horizons, there is ically given spacetime the constraints are only satisfied no reason why apparent horizons should deform or join up to the discretisation error. It is thus in my opinion a when two black holes approach each other. bad idea to calculate the ADM mass (which is a global Figure 6 shows the the common apparent horizons for quantity) from information near the outer boundary only, a series of black holes with increasing distances. As the relying on the constraints to propagate the necessary in- black holes are placed further apart, the common hori- formation to the boundary. A better method to calculate zons become unsurprisingly more elongated. Figure 7 the ADM mass is needed. I decided to present the ADM shows both the individual and common black holes for values here nonetheless, because they (although inaccu- varying distances for comparison. Near z = 3.0, the rate) still provide some insight into the spacetimes. common horizon ceases to exist. ± 8

IX. SUMMARY z=0.0 4 z=1.0 z=2.0 z=2.5 z=3.0 2 Apparent horizons provide valuable insight into space- times, such as mass and spin estimates for black holes.

x 0 They are indicators for event horizons, and help keep track of the location of singularities. They provide nec- essary information for the time evolution of black hole -2 spacetimes. This apparent horizon finding algorithm pro- vides a fast way to track horizons during a numerical evo-

-4 lution, and also a robust way to find horizons in initial data, or as they appear during an evolution. -6 -4 -2 0 2 4 6 8 z

Figure 6: Common horizons for a series of merged black holes with increasing distances.

4 z=2.0 z=3.0 3 z=4.0

2

1 Acknowledgements

x 0

-1

-2 I wish to thank to my collaborators Mijan Huq, Badri -3 Krishnan, Pablo Laguna, and Deirdre Shoemaker for countless inspiring discussions. Hans-Peter Nollert pro- -4 -6 -4 -2 0 2 4 6 vided helpful comments on this manuscript. While work- z ing at this project, I was financed by the SFB 382 Ver- ” fahren und Algorithmen zur Simulation physikalischer Figure 7: Individual and common horizons for superposed Prozesse auf H¨ochstleistungsrechnern“ of the DFG. Sev- Kerr–Schild black holes near z = ±3.0, where the common eral visits to Penn State University were supported by apparent horizon disappears. the NSF grants PHY-9800973 and PHY-0114375.

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