Curvature Invariants and the Geometric Horizon Conjecture in a Binary Black Hole Merger
CCurvature Invariants and the Geometric HorizonConjecture in a Binary Black Hole Merger
Jeremy M. Peters , Alan Coley , and ErikSchnetter
Department of Mathematics and Statistics, Dalhousie University,Halifax, Nova Scotia, B3H 3J5, Canada Perimeter Institute for Theoretical Physics, Waterloo, Ontario,N2L 2Y5, Canada Department of Physics & Astronomy, University of Waterloo,Waterloo, ON N2L 3G1, Canada Center for Computation & Technology, Louisiana StateUniversity, Baton Rouge, LA 70803, USAJanuary 26, 2021
Abstract
We study curvature invariants in a binary black hole merger. It hasbeen conjectured that one could define a quasi-local and foliation indepen-dent black hole horizon by finding the level–0 set of a suitable curvatureinvariant of the Riemann tensor. The conjecture is the geometric horizonconjecture and the associated horizon is the geometric horizon. We studythis conjecture by tracing the level–0 set of the complex scalar polynomialinvariant, D , through a quasi-circular binary black hole merger. We ap-proximate these level–0 sets of D with level– ε sets of |D| for small ε . Welocate the local minima of |D| and find that the positions of these localminima correspond closely to the level– ε sets of |D| and we also comparewith the level–0 sets of Re( D ). The analysis provides evidence that thelevel– ε sets track a unique geometric horizon. By studying the behaviourof the zero sets of Re( D ) and Im( D ) and also by studying the MOTSs andapparent horizons of the initial black holes, we observe that the level– ε set that best approximates the geometric horizon is given by ε = 10 − . Black holes are solutions of general relativity and are most naturally character-ized by their event horizon. The event horizon of a black hole (BH) is definedas the boundary of the causal past of future null infinity. Intuitively, this meansthat on the inner side of the event horizon, light cannot escape to null infinity.1 a r X i v : . [ g r- q c ] J a n Notice that event horizons require knowledge of the global structure of space-time [8, 9, 10]. However, for numerical relativity it is more convenient to usean initial value formulation of GR (a 3+1 approach), where initial data is givenon a Cauchy hypersurface and is then evolved forward in time. This approachrequires an alternative description of BH horizons which is not dependent of theBH’s future. [56, 55, 3, 4, 31, 28].Let Σ be a compact 2D surface without border and of spherical topology, andconsider light rays leaving and entering Σ, with directions l and n , respectively.Let q ab be the induced metric on Σ and denote the respective expansions asΘ ( l ) = q ab ∇ a l b and Θ ( n ) = q ab ∇ a n b [50]. Then, Θ ( l ) and Θ ( n ) are quantitieswhich are positive if the light rays locally diverge, and negative if the light rayslocally converge, and are zero if the light rays are locally parallel. We say thatΣ is a trapping surface if Θ ( l ) < ( n ) < ( l ) = 0 [48, 45, 46, 52, 26, 29]. (Σ is a future MOTS if Θ ( l ) = 0 andΘ ( n ) < ( l ) = 0 and Θ ( n ) > q ab restricted to the spacetime hypersurface and the extrinsic curvature of thathypersurface at a given time [27, 10, 26]. Gravitational fields at the AH arecorrelated with gravitational wave signals [27, 34, 33, 30, 29, 50], so AHs areuseful to study gravitational waves. AHs are also used to numerically simulatebinary black hole (BBH) mergers and the collapse of a star to form a BH [13].For example, AHs play a role in checking initial parameters and reading offfinal parameters of Kerr black holes in gravitational wave simulations at LIGO[13, 2, 1]. DHs are also useful, as they could contribute to our understanding ofBH formation [8, 9, 10, 13]. In addition, MOTSs turn out to be well-behavednumerically, and also can be used to trace physical quantities of a BH as theyevolve over time and through a BBH merger [52, 26, 29, 48, 49]. One possibledisadvantage of AHs is that the definition of AHs as the ”outermost MOTS” isonly useful in practice when a foliation is given [7].It has been conjectured that one can uniquely define a smooth, locally deter-mined and foliation invariant horizon based on the algebraic (Petrov) classifica-tion of the Weyl tensor [22, 21]. The necessary conditions for the Weyl tensor tobe of a certain Petrov type can be stated in terms of scalar polynomials in theRiemann tensor and its contractions which are called scalar polynomial (cur-vature) invariants (SPIs). The first aim of this work is to study certain SPIsnumerically during a BBH merger.The Petrov classification is an eigenvalue classification of the Weyl tensor,valid in 4 dimensions (D). Based on this classification, there are six differentPetrov types for the Weyl tensor in 4D: types I , II , D , III , N and O (which isflat spacetime). One can also use the boost weight decomposition to classify theWeyl tensor, which is equivalent in 4D to the Petrov classification. One can alsoalgebraically classify the symmetric trace free operator, S ab , that is the tracefree Ricci tensor, which is equivalent to the Segre classification [54].The boost weight algebraic classification generalizes the Petrov classificationto N -dimensional spacetimes [22, 21, 23, 24, 15, 40]. In N D, we start with theframe of N –vectors, { l , n , { m i } N − i =2 } , where l and n are null, l · n = 1, andthe { m i } are real, spacelike, mutually orthonormal, and span the orthogonalcomplement to the plane spanned by l and n . The possible orthochronousLorentz transformations are generated by null rotations about l , null rotationsabout n , spins (which involve rotations about m i ), and boosts [40]. With respectto the given frame, boosts are given by the transformation: l → λ ln → λ − nm i → m i for all i ∈ { , . . . , N − } and for some λ ∈ R \{ } . (The remaining transforma-tions are given in [23, 24, 15, 40].) It is possible to decompose the Weyl tensorinto components organized by boost weight [23, 24, 15].It is of particular interest to know whether a given 4D spacetime is of specialalgebraic type II or D . We can state the necessary conditions as discriminantconditions in terms of simple SP I s [22, 21, 16, 18]. Just as an
SP I is a scalarobtained from a polynomial in the Riemann tensor and its contractions [22, 21],an
SP I of order k is a scalar given as a polynomial in various contractions ofthe Riemann tensor and its covariant derivatives up to order k [22, 21]. It turnsout that BH spacetimes are completely characterized by their SP I s [17]. Thenecessary discriminant conditions on the 4D Weyl tensor for the spacetime tobe of type II / D can be stated as two real conditions and are given in [17].Contracting the 4D (complex) null tetrad, ( l , n , m , m ) where m and m arecomplex conjugates, with the Weyl tensor, C abcd , one may form the complexscalars, Ψ , Ψ , Ψ , Ψ , Ψ and, in terms of these scalars, as in the Newman-Penrose (NP) formalism (discussed later), one may define the scalar invariants: I = Ψ Ψ − Ψ Ψ + 3 Ψ (1) J = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Ψ Ψ Ψ Ψ Ψ Ψ Ψ Ψ Ψ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (2)It can be shown that the two aforementioned real scalar conditions are equivalentto the real and imaginary parts of the following complex syzygy [54]: D ≡ I − J = 0 (3)Thus for Petrov types II and D , equation (3) holds everywhere. It also turnsout that for Petrov types III , N , and for O , we have I = J = 0, so (3) issatisfied trivially. Having discussed the Petrov and boost weight classifications, we now turn to theGeometric Horizon Conjecture (GHC) in which we define the geometric horizon(GH) as the set on which the
SP I s, defined in (3), vanish [22, 21]. The level–0 sets of these
SP I s might not form a horizon with nice properties, however,since these
SP I s could vanish additionally on axes of symmetry or fixed pointsof isometries [22, 21]. We know from (3) that if the spacetime is algebraicallyspecial, then the given complex
SP I vanishes. More precisely, the GHC is givenas follows [22, 21]:
GH Conjecture:
If a BH spacetime is zeroth-order algebraically general, thenon the geometric horizon the spacetime is algebraically special. We can identifythis geometric horizon using scalar curvature invariants.
Comments:
Note that when studying the GHC, one would need to ensurethat the GH exists and is unique. If the GHC were true in an algebraicallygeneral spacetime, then one could say on this horizon, the Weyl tensor is morealgebraically special than its background spacetime and this horizon is at leastof type II . This horizon is then foliation independent and quasi-local [22, 21].If the spacetime is algebraically special, one then considers the second partof the GHC, which is analogous to the algebraic GHC above, but involving dif-ferential SP I s . Differential SP I s (of order k ≥
1) are scalars obtained frompolynomials in the Riemann tensor and its covariant derivatives and their con-tractions. This second part of the GHC thus states that if a BH spacetimeis algebraically special (so that on any GH the BH spacetime is automaticallyalgebraically special), and if the first covariant derivative of the Weyl tensoris algebraically general, then on the GH the covariant derivative of the Weyltensor is algebraically special, and this GH can also be identified as the level–0set of certain differential
SP I s [22, 21]. In this case, the GH is identified asthe set of points on which the covariant derivative of the Weyl tensor, C abcd ; e is of type II [22]. It follows that one may obtain a clearer picture of the GH bytaking the level–0 sets of these differential SP I s.In addition to
SP I s, Cartan invariants can play a role within the frameapproach and they are easier to compute. For example, Cartan invariants areuseful in event horizon detection; indeed, it was demonstrated in [39, 20] that in4D and 5D, one can construct invariants in terms of the Cartan invariants whichdetect the event horizon of any stationary asymptotically flat BH solutions. Onecould rewrite the statement of the algebraic GHC in the language of the boost-weight classification [40] to say that ”if there is some frame with respect towhich the Weyl tensor in a BH spacetime has a vanishing boost-weight +2term, then on the GH, there is some frame with respect to which the Weyltensor has a vanishing boost-weight +1 term.” This desired frame is calledthe algebraically preferred null frame (APNF). For example, in an algebraicallygeneral 4D spacetime, the APNF is the frame in which the Weyl tensor is ofalgebraic type I so that the boost weight +2 terms of the Weyl tensor are 0 withrespect to this frame, which is always possible in 4D. Then, the GH is identifiedas the set of points on which the terms of boost weight +1 are zero. (If theWeyl tensor is type II , then one can analyze the covariant derivative of theWeyl tensor and ask for it to be algebraically special). The task in this frameapproach to study the GHC, therefore, is to first find this APNF, ( l, n, m, m )and thus the orthogonal AHs/DHs [53]. To this end, the Cartan algorithm canbe used to completely fix this frame [39], and with respect to this frame, oneobtains the associated Cartan scalars. From these scalars, one can identify thelevel–0 set of C abcd ; e with the GH and, via NP calculus, obtain the NP spinexpansion coefficients with respect to this APNF. It is of particular interest tostudy the spin coefficients, ρ and µ , as their level–0 set could be associatedwith the GH. This definition is related to that of AHs and DHs, but with theadditional geometric/algebraic motivation afforded by the Cartan scalars. Thereare many examples, outlined in the next section, which motivate identifying theGH as the level–0 set of ρ and µ , but a more careful study of ρ , µ and theirevolution through a BBH merger is beyond the scope of this paper.In this paper, we shall study the (algebraic) SP I s in relation to the first(algebraic) part of the GHC. More specifically, we will study the complex level–zero set of the invariant, D = I − J , as given in (3), in 4D during a BBHmerger. This could possibly help provide insight as to whether one can define aproper unique horizon based on the algebraic classification of the Weyl tensor.This conjecture might have to be modified so that instead of analyzing level–0 sets of the real SP I s, we analyze instead level– ε sets for small ε . Such an ε could be determined by locating the local minima of the SP I s. However,further evidence from the analysis of D r below perhaps suggests that this is notthe case. There are many examples to support the plausibility of the GHC [39, 20, 8, 9, 10].For example, in the Kerr spacetime, by invoking the notion of a non-expandingweakly isolated null horizon and an isolated horizon, it can be proven, usingthe induced metric and induced covariant derivatives on the submanifold andassuming the dominant energy condition, that the Weyl and Ricci tensors areboth of type II / D on the event horizon. This means that one can extracta subset of the set of points where the Weyl and Ricci tensors are both ofalgebraic type II / D , to define a smooth BH horizon, namely the event horizon[5, 36, 8, 9, 10, 22, 21]. It can also be shown that the covariant derivatives of theRiemann tensor are also of type II on this horizon [22, 21]. The Kerr geometrycan be approximated by the spacetime of a BH formed by a collapsing star.Thus, there should be a horizon surrounding the event horizon for a collapsingBH that can be identified using the algebraic conditions on the Riemann tensormentioned earlier. By continuity, the inside of the event horizon should alsoapproximate the Kerr geometry, and the Kerr geometry admits an inner horizon.This inner horizon is shown to be a null surface, but is unstable, allowing forthe possibility that the GH is unique at later times [8, 9, 10, 5, 36, 22, 21].Another example to support the GHC comes from a family of exact closeduniverse solutions to the Einstein-Maxwell equations with a cosmological con-stant representing an arbitrary number of BHs, discovered by Kastor and Traschen(KT) [35]. Consider the merger of 2 BHs. At early times, there are two 3D dis-joint GHs forming around each BH [22, 21]. However, at intermediate times,it turns out that the invariant, D = I − J = 0 as in (3) only at the co-ordinate positions of each of the BHs, along certain segments of the symmetryaxis, and along a 2D cylindrical surface, which expands to engulf the 2 BHs asthey coalesce [22, 21]. During the intermediate process, there are 3D surfaceslocated at a finite distance from the axis of symmetry for which the tracelessRicci tensor (and hence the Ricci tensor, R ab ) is of algebraic type II / D . Thereis also evidence of a minimal 3D dynamically evolving surface where a scalarinvariant, W , assumes a constant minimum value. This suggests that there isa GH during the dynamical regime between the spacetimes [22, 21], but furtherinvestigation is needed. At late times, the spacetime then settles down to atype D Reissner-Nordstrom-de-Sitter BH spacetime with mass M = m + m ,which turns out to have a GH [39, 20]. So a GH forms at the beginning and endof the coalescence. For further information on the two-BH solution, see [35].The KT solution for multiple BHs was studied and GHs around each BH werefound in [38]. The results were compared with the previously mentioned 2-BHsolution. Additionally, three black-hole solutions were studied and GHs werefound around these BHs also [22, 21]. For information on more than two BHs,see [42].There are additional examples of spacetimes that support the GHC either byexplicitly exhibiting GHs or by finding other established BH horizons on whichthe Weyl and Ricci tensor are algebraically special [22, 21, 19, 54]. One canalso verify the GHC by identifying GHs with the level–0 sets of the NP spincoefficients, ρ and µ , as mentioned before. There are four examples to supportthis. The first example is in stationary spacetimes with stationary horizons (e.g.Kerr-Newman-NUT-AdS) [39]. In this spacetime, the Weyl and Ricci tensorsare type D everywhere, and the APNF can be chosed to make this conditionmanifest. Here, the GH can be identified as the set on which the covariantderivatives of the Weyl and Ricci tensors are of type II , and this coincideswith the level–0 set of ρ [39]. The second example is in spherically symmetricspacetimes such as vacuum solutions or known exact solutions (e.g. Vaidya orLTB dust models) [22]. In this case, the Weyl tensor is of type D and here theRicci tensor provides no useful information. Adopting the APNF to the type- D condition of the Weyl tensor, we then find that the GH coincides with the set ofpoints on which the covariant derivative of the Weyl tensor is of type II , whichis also the level–0 set of ρ . For the third example, consider quasi-sphericalSzekeres spacetimes [19], where the Weyl tensor is of type D and the Riccitensor is of type I . Studying the components of the covariant derivative of theWeyl tensor, shows that ρ = 0 precisely on the GH. A similar situation holdsin the fourth example [38], which is the Kastor-Traschen solution for N > I and the Ricci tensor is of type D . Withrespect to an adapted APNF, it follows that ρ = 0 and µ = 0 precisely on the set { C abcd ; e is of type D } , which identifies the GH. The authors are also currentlystudying vacuum solutions in the case of axisymmetry (so R ab = 0) and wherethe Weyl tensor is of algebraic type I . Based on the previous examples, it isnatural to study the covariant derivative of the Weyl tensor in this setting andit is also of interest to study the level–0 sets of the NP spin coefficients ρ and µ . We wish to study the behaviour of the complex
SP I , D , as defined in (3),through a BBH merger. Since the Kerr geometry is type D everywhere, itfollows that D = 0 everywhere for a Kerr BH. It is known that in a BBH mergerthe merged BHs at late times settle down to a solution well described by theKerr metric [22, 21]. Thus, for a merger of 2 initially Kerr BHs, a plot of the realpart and imaginary part of D should be roughly zero everywhere at early andlate times. However, in the intermediate “dynamical” region (during the actualFigure 1: Contour plots of |D| during a quasi-circular BBH merger consisting oftwo merging, equal mass and non-spinning BHs at selected times t = 8 (upperleft), t = 12 (upper right), t = 16 (middle left), t = 18 (middle right), t = 20(lower left) and t = 24 (lower right).merger and coalescence at intermediate times), these same zero plots shouldyield important information. This is what we wish to study.We highlight some known features of a binary black hole merger, as describedin [48, 49, 27]. This also serves to set up our notation. In [48, 49], it was foundthat there is a connected sequence of MOTSs, which interpolate between theinitial and final states of the merger (two separate BHs to one BH, respectively)[48, 49]. The dynamics are as follows: Initially, there are two BHs with disjointMOTS (which are AHs at this point [27]), S , and S , one around each BH.Then, as the two BHs evolve, a common MOTS forms around the two separateBHs and bifurcates into an inner MOTS, S i , which surrounds the MOTS andan outer MOTS, S c . S c increases in area and encloses S , S and S i , and is theAH at the time of the merger [27, 48, 49]. The fate of S c and the bifurcation atthe time of the merger is well understood [27, 9, 52, 6, 29, 48, 49, 32, 41, 47].Figure 2: Comparing selected local minima of |D| along the x–coordinate direc-tion with selected level sets of |D| at times t = 12 (upper plot), t = 16 (middleplot) and t = 20 (lower plot).Figure 3: Comparing level– ± D r = Re( D ) and D i = Im( D ) withlevel—0 .
001 contours of |D| . The upper, middle and lower left plots are plotsof D r = Re( D ) at times t = 12 , ,
20, respectively and the upper, middle andlower right plots are plots of D i = Im( D ) at times t = 12 , ,
20, respectively.0Figure 4: Comparing the white level–0 .
001 sets of |D| with the MOTS as de-scribed in [48, 49] at times t = 12 (upper left), t = 16 (upper right) and t = 20(lower left and right). The lower left panel shows the inner MOTS in purplewhereas the lower right panel shows the outer MOTS in purple.1 Instead of studying a head-on collision, in this paper we shall study a quasi-circular orbit of two merging, equal mass and non-spinning BHs. This simulationhas not been presented elsewhere and the results of this simulation are new.In these simulations, the Einstein toolkit infrastructure was used [37] and thesimulations are run using 4 th order finite differencing on an adaptive mesh grid,with adaptive refinement level of 6 [51, 12]. Brill-Lindquist initial data withBH positions and momenta set up to satisfy the QC-0 initial condition wereused [25]. Instead of analyzing a sequence of MOTS throughout the merger, weseek to define and study a GH as it evolves through the merger, in accordancewith the GHC. Since (3) sets necessary conditions for the Weyl tensor to beof algebraic type II , we seek to analyze the constant contours of the difference D = I − J . In the simulations, the real and imaginary parts of I and J arecalculated using the Cartan invariants, { Ψ i } i =0 , as given in equations (1) and(2), and the calculations are carried out using the orthonormal fiducial tetrad,as given by [11].Note that in the rest of this paper, we will use the notation from [48, 49, 27]to describe the various MOTSs that appear in our simulations. To recapitulate, S and S are the 1st and 2nd initial MOTS and S i and S c describe the innerand common MOTS as they appear in our simulation, respectively. We will alsobe interested in the spherical approximations of S and S . Let S be a sphericalsurface centred at the centroid of S and whose radius is the average radius of S at each time during the merger. Correspondingly, let S be a spherical surfacecentred at the centroid of S whose radius is the average radius of S . We willalso make use of the spherical surface, S c , which is centred at the centroid of S c whose radius is the average radius of S c .Figures 1-4 provide plots of various level sets of D r = Re { D } , D i = Im { D } and |D| as functions of ( x, y ) ∈ R at a fixed spatial coordinate value of z =0 . t , where t = 0 indicatesthe start of the numerical computation. These level sets are also compared with S and S . The full compliment of pictures describing this BBH merger aredisplayed in [44]. We present a subset of those figures to illustrate the essentialfeatures. In each of Figures 1-4, the data corresponding to x < x > x = y = 0axis. In Figures 1–4, we plot the centroids and outlines of S and S along with S and S , and also S i and S c , when they have formed. It is found that S and S provide valid spherical approximations to S and S . Figure 1 provides the contour plots of the magnitude of D = I − J , denoted |D| , on a log scale (see (3)) for t = 8 , , , , ,
24 in the upper left, upperright, middle left, middle right, lower left and lower right panels, respectively,for fixed z = 0 . |D| is positive definite, the level–0 sets of |D| are impossible to find precisely due to discrete resolution and numerical error.Instead, we highlight the evolution of the level– ε sets, where ε = 3 × − , × − , × − . The overlaid green, red and white contours of each frame arethe level–3 × − , level–5 × − and level–1 × − sets of |D| , respectively.The blue dots in Figure 1 give the centres of S and S (or, in other words, the2centroids of S and S ) as they evolve and serve to track the positions of thecorresponding initial BHs through the merging process. [44] has plots of |D| withthe relevant level– ε sets superimposed at all times t = 0 , . . . , , , , t = 8 and t = 12 in the upper left and right panels,respectively), each of the level– ε sets are partitioned into pairs of simple closedcurves, each of which contains the centroids of S and S , respectively. At t = 16(middle left panel), the red level–5 × − set and the white level–1 × − setof |D| each form a third simple closed curve between S and S and this thirdsimple closed curve is centred at ( x, y ) = (0 , × − setforms its third simple closed curve at time t = 14, but this is not shown in thepaper for brevity (see [44] for details)). At times t = 18 (middle right panel), t = 20 (bottom left panel) and t = 24 (bottom right panel), for each respective ε = 3 × − , × − , × − , the multiple simple closed curves partitioningthe level– ε set of |D| have joined so that each level– ε curve is now a singlesimple closed curve surrounding the 2 centroids of S and S . It follows thatthe level– ε curves for each ε = 3 × − , × − , × − at each t form aninvariantly defined, foliation invariant horizon that contains each separate BHat early times, and contains the merged BH at late times.The evolution of the level– ε curves through the quasi-circular BBH mergerin Figure 1 is reminiscent of the sequence of MOTS that take place during thehead-on collision simulation in [48, 49]. In particular, in [48 ,
49] after the twoseparate initial BHs start to merge together, a third MOTS forms and bifurcatesinto S i and S c . This bifurcation, also summarized in [27], can be compared toour quasi-circular BBH merger simulations, particularly at t = 16 in the middleleft panel of Figure 1 when each level– ε set is partitioned into three simpleclosed curves for ε = 3 × − , × − , × − . However our numericalcomputations are not precise enough to study the details of the bifurcation, asfound in [48, 49, 27]. At time t = 24, in the lower right panel of Figure 1, it alsoseems that the centroids of S and S do not merge fully. Thus, it is possiblethat at at late times, the level– ε sets of |D| for ε = 3 × − , × − , × − may track S and S , which have been found in [27] to overlap but not intersectat late time. However, our simulations did not run to late enough times to makethis clear.Consequently, Figure 1 provides strong evidence that for each ε = 3 × − , × − , × − , the level– ε sets of |D| track a unique GH, whichcan be identified by the level–0 set of D . In [44], we also found that a sub-set of the level–0 sets of D r and D i both track the level– ε sets of |D| for all ε = 3 × − , × − , × − at all times t = 0 , . . . , , , ε sets of |D| approximatethe level–0 sets of D . From Figure 1, we have evidence of the fact that thelevel– ε sets track the GHs through all stages of the BBH merger, including thetime when the level– ε sets of |D| start as disjoint simple closed curves with eachsurrounding the centroids of S and S , respectively, when the level– ε sets of |D| are partitioned into three simple closed curves at intermediate times, andwhen the level– ε sets of |D| form a single simple closed curve surrounding bothcentroids of S and S at late times.While Figure 1 shows that the level– ε sets of |D| are well behaved as GHs,the GHC implies the existence of a horizon on which the Riemann tensor is alge-braically special which, in turn, implies the existence of a horizon characterizedby the level–0 set of D . One has by definition that the complex invariant, D = 03if and only if its magnitude, | D | = 0. However, numerically finding the level–0sets of |D| is extremely difficult because |D| is a positive definite quantity, sothat any numerical errors would provide a positive contribution. Furthermore,in the numerical simulation it is possible that the actual zeros of |D| do notoccur at any points which are sampled for the discrete mesh being used. Thus,the level– ε sets of |D| could indeed approximate the level–0 sets of the complexinvariant, D . It remains to estimate the appropriate preferred value for ε .We observe that for ε = 3 × − , × − , × − , the level– ε contoursare very close to each other, showing that the level– ε sets vary continuously with ε . We also observe that if ε ≤ ε , then the 2D area enclosed by the level– ε curve encloses the 2D area enclosed by the level– ε curve. Thus, each panelof Figure 1 indicates that |D| decreases on average with average distance fromthe centroids of S and S , which means that the plots of |D| show no globalminima. However, Figures 2 and 3 indicate that the plots of |D| do have havelocal minima which approximately coincide with the level–0 sets of D r and withthe level– ε sets for ε = 3 × − , × − .In order to investigate further the level–0 sets of |D| (or equivalently thelevel–0 sets of D ), we find out where |D| takes a local minimum value. If thevalue of |D| itself is small, then these locations of the local minima could possiblyindicate positions of the actual zeros of |D| , which would be caused by numericalerrors. For example, the zero of |D| could be “missed by the discretization”–i.e. the contour plots of |D| could display local minimum values on the meshwhen |D| is theoretically zero. It could also be the case that the GHC should bemodified so that the GH is defined as the set of points where |D| reaches the localminimum instead of being identically zero. If this were the case, then locatingthe local minima of |D| would locate the GH precisely instead of approximatingit. However, further evidence from the analysis of D r below perhaps suggeststhat this is not the case.To this end, we examine 1D plots of |D| as functions of y for fixed x , hence-forth referred to as “slice plots.” Along each slice plot, we find the values of y = y min , where |D| assumes a local minimum value, and record the cor-responding point ( x min , y min ). In [44], we show explicitly the slice plots ofthe contour plots of |D| at times t = 12 , ,
20 and demonstrate explic-itly the process of finding the local minima of |D| . In Figure 2, we restrictour attention to finding the positions of the local minima of |D| whose cor-responding |D| values lie within the range (cid:2) × − , . × − (cid:3) which con-tains (cid:8) × − , × − , × − (cid:9) , the set of values of ε being consideredfor level– ε sets of |D| . The positions of the local minima of |D| correspondingto this desired range are plotted with green dots in Figure 2. The remainingfeatures of Figure 2 are the same as in Figure 1.In summary, Figure 2 gives the contour plots of |D| at times t = 12 , , × − sets of |D| , white level–1 × − sets of |D| and the blue points marking thecentroids of S and S , as in Figure 1. The points where |D| assumes a localminimum value and lies in the range (cid:2) × − , . × − (cid:3) are plotted usinggreen dots. [44] also presents these same plots of |D| , along with the green localminima, at magnified resolution.Technically speaking, the green points in Figure 2 correspond to local min-ima of |D| only along slice plots of |D| vs y for a fixed x . In [44], we have also4examined 1D plots of |D| as functions of x for fixed y and found that the setof points, ( x (cid:48) min , y (cid:48) min ) ∈ R where |D| assumes a local minimum value alongeach fixed y = y (cid:48) min “slice plot” corresponds closely with the set of points,( x min , y min ) ∈ R where |D| assumes a local minimum value along each fixed x = x min slice plot. Thus, the points where |D| assumes a local minimum valuealong its corresponding slice plot accurately approximate the points correspond-ing to an overall local minima of |D| .We observe that at times t = 12 and t = 16, these local minima appear totrack the green level–3 × − sets, while at time t = 20, these local minimaappear to track more closely the white level–1 × − sets. In any case, thepositions of the local minima of |D| align closely with the level- ε sets of |D| for ε = 3 × − , × − . This is also described in [44] in more detail. Therefore,Figure 2 demonstrates that positions of the local minima of |D| accurately trackthe GH and provides supporting evidence that the level– ε sets track the GH for ε = 3 × − , × − . In [44], we also track the level–0 sets of D r and comparewith the locations of the local minima of |D| at original resolution, and also findthat all local minima of |D| track closely the level–0 sets of D r .The problem of estimating the level–0 sets of the complex D cannot bedefinitively resolved by analyzing |D| because, as previously noted, |D| is apositive definite quantity and the discrete resolution imposed by the numericalsimulation does not necessarily allow one to accurately find level–0 sets of |D| .Thus, to gain further insight into the GH through the BBH merger, it is thereforehelpful to analyze quantities which change sign through a zero. By continuity,the level–0 sets of these quantities can be deduced from adjacent positive andnegative level– ε sets for small ε . This avoids some of the problems of numericalresolution that occur when using the local minima of |D| to estimate the zerosof |D| (and hence D ). The quantities we choose to analyze are D r = Re( D ) and D i = Im( D ).In Figure 3 we plot, with magnified resolution, the contour plots of D r (left panels) and D i (right panels) along with their level– − .
01 sets in yellowand their level–+0 .
01 sets in lime green in contours of D r (left panels) and D i (right panels) at times t = 12 , ,
20 in the upper, middle and lower panels,respectively. In [44], we also analyzed the quantities Re( D ) = D r − D i andIm( D ) = 2 ∗ D r ∗ D i with their associated level– ± .
01 sets, in addition to thelevel– ± .
01 sets of D r and D i . The grey regions in each of the frames in Figure3 correspond to regions where − . < D r < .
01 (resp. − . < D i < . D r ≥ D i ≥
1) and the white regions correspond to regions where D r ≤ − D i ≤ − S and S .Furthermore, in Figure 3 (and in the relevant figures in [44]), we comparedthe level– ± .
01 sets of D r and D i with the white level–1 × − sets of |D| ofFigures 1–2. We know that a positive level set and a nearby negative level setindicates a sign change of the quantity being plotted. Hence, there must be asurface among the level ± .
01 sets of D r (resp. D i ) across which D r (resp. D i )change sign. This surface gives the level–0 set of D r (resp. D i ). By estimatingthe zeros of D r or D i in this manner, we reduce the possibility of numerical noisethat comes with plotting the level–0 sets of D r or D i directly. Upon inspectionof each frame in Figure 3, we see that the level– ± .
01 sets of D r (resp. D i ) occurin close proximity with but are contained in the interior of the level–1 × − sets5of |D| . Thus, each of D r and D i have level–0 sets which occur in close proximityto the level– ± .
01 sets of D r and D i , respectively, and closely approximate thelevel–1 × − sets of |D| . It follows that an examination of the contours of D r and D i in Figure 3 provides strong evidence that the level–1 × − set of |D| well approximates the elusive level–0 sets of the complex invariant, D .We next explicitly compare the level–1 × − sets of |D| with the corre-sponding MOTSs S , ,i,c and the spherical approximations, S , ,c , as describedbelow, in Figure 4. We display the 2D contour plots of |D| with magnified res-olution at times t = 12 and t = 16 in the upper left and upper right panels,respectively, and we display the 2D contour plots of |D| at t = 20 and in thelower left and lower right panels. As in Figures 1–3, the white curves denotethe white level–1 × − sets of |D| . The dark blue ellipses in all frames trackthe outline of the ( x, y ) coordinates of a z = 0 . S ∪ S andthe blue points in all frames of Figure 4 mark the centroids of S and S , asbefore. The light sky blue curves mark the ( x, y ) coordinates of points on S ∪S whose corresponding z coordinate values lie in the range [0 . , . S and S are plotted with light sky blue dots. Bydefinition, the centroids of S and S align exactly with the centroids of S and S , respectively.In the present quasi-circular simulation, the bifurcation of the the thirdMOTS into S i and S c occurs between times t = 18 . t = 18 .
75. Oncethis happens, S and S are no longer AHs, as the MOTS, S c , now surrounds S , S and S c . Thus, in order to compare our level– ε sets of |D| with AHs(the outermost MOTSs), we have included plots of S i and S c here. The purpledots on the bottom left (resp. bottom right) panel of Figure 4 label the ( x, y )coordinates of the points on S i (resp. S c ) whose corresponding z value lies inthe range, [ − . , +0 . z = 0 . S c . Note that in the bottom right–hand corner, the outer MOTS atlate times is so big that the entire two dots and the scale for scale of the whitelevel–0 .
001 sets of |D| and S i, , are squashed at the origin.From each panel of Figure 4, we notice that the MOTSs, S and S , areclosely mimicked by S and S , respectively. Hence, S and S are nearlyspherically symmetric surfaces. We also find that the white level–1 × − setof |D| coincides closely with S and S , especially at early times. Since theMOTSs S c and S i have not yet formed at such early times, S and S are AHsat early times. Therefore, Figure 4 shows us that the white level–1 × − sets of |D| well approximate the AH at early times. At later times, however, itappears that the AH, S c , diverges from the white level–1 × − sets, so that thewhite level–1 × − sets of |D| no longer approximate the AH in this regime.(Note that this AH, S c , is well approximated by its spherical approximation, S c .) Instead, these white level–1 × − sets of |D| are approximated reasonablyclosely by S i and especially S and S . This lends support to the choice of thewhite level–1 × − sets of |D| as a representative approximation to the level–0set of |D| (and hence of D ).In [44], we compare S and S to the level–5 × − sets of |D| at all times t = 0 , . . . ,
26 and find that these red level–5 × − sets track the sphericallyaveraged AH extremely closely, especially at early times. Since the white level–1 × − sets and red level–5 × − sets track each other extremely closely,this lends support to the fact that the white level–1 × − sets of |D| are well6approximated by the AHs of the initial BHs at all times. We presently choosethe white level–1 × − set of |D| as our representative level– ε set of |D| toapproximate the level–0 set of |D| and thus the level–0 set of D . In [44], wealso have studied the contour plots of D r and D i and with their level–0 sets atall times t = 0 , . . . , , , ,
42. Here we have illustrated the plot featurespresent in Figures 2–4 in [44] for times t = 12 , ,
20. The simulation whosefigures we present in [44] has not been presented elsewhere and the results ofthis simulation are new.Therefore, Figures 1–4 provide strong evidence that one can define a uniquesmooth GH, theoretically given by the level–0 set of the complex invariant D = I − J , which we have found is best approximated in the numerics by thelevel–1 × − sets of |D| . We have studied the algebraic properties of the Weyl tensor by analyzing thetime evolution of various level– ε sets of |D| through a quasi-circular merger oftwo non-spinning, equal mass BHs where, in particular, ε = 3 × − , × − , × − . These level– ε contours are superimposed on the contour plotsof |D| in Figure 1. In these plots, the locations of the two initial BHs weretracked by using the centroids of the initial AHs. We found that at early times,each such level set is partitioned into two disjoint simple closed curves, eachof which contains one of the two centroids of the AHs of the 2 separate initialBHs. Then each level set, at some intermediate time, forms a third simple closedcurve which is centred at the origin and positioned between the centroids of theAHs of the two initial BHs. These three simple closed curves then join and formone simple closed curve for each level set, which contains the centroids of bothinitial BHs.The plots for |D| in Figure 1 provide strong evidence that the level sets of |D| identify the GH. However, it is impossible to identify the level–0 sets of |D| precisely, since |D| is a sum of positive definite terms, so numerical errorsand discrete resolution cause |D| to be strictly positive. Thus, to further studythe zeros of |D| , which would indicate the zeros of the complex quantity D , westudied the positions of the local minima of |D| along “slice plots” of |D| vs y for a fixed x in Figure 2. Figure 2 demonstrates that the level– ε sets of |D| correspond closely to the local minima of |D| , where ε = 3 × − , × − .Since the local minima of |D| approximate the zeros of |D| , Figure 2 providessupporting evidence that the level– ε sets of |D| for ε = 3 × − , × − trackthe GH of the BBH merger.Since |D| is positive definite, its zeros cannot be traced by positive and neg-ative level sets. Therefore, we have also analyzed quantities which change signthrough a zero. In Figure 3, we examined contour plots of D r = Re( D ) and D i = Im( D ) with their associated level– ± .
01 sets and compared these plotswith the white level–1 × − sets of |D| . We found surfaces surrounding theunion of the level– ± .
01 contours of D r (resp. D i ) across which D r (resp. D i )change sign, and are hence a subset of the level–0 sets of D r (resp. D i ). Wealso found these particular zeros of D r (and those of D i ) to be well approxi-mated by the level–1 × − contours of |D| in our plots and suggest that thisapproximation is valid in this setting.7In Figure 4, we compare the level–1 × − contours of |D| with the AHs.These AHs are given by S and S at early times and by S c after S c has formed.Figure 4 shows that S and S are reasonably approximated as spherically sym-metric surfaces at all times. We also find that the level–1 × − sets of |D| are very well approximated by the AHs, S and S , at early times, but at latertimes the AH diverges from this level set of |D| . However, even at late times, S and S continue to track a subset of the level–1 × − set of |D| , as does S i . Therefore, in the binary black hole merger, as displayed in Figures 1–43 in[44] and summarized in Figures 1–4 above, the algebraic structure of the Weyltensor is clearly identified by the level– ε sets of |D| , and it is plausible that thelevel set with ε = 1 × − accurately identifies the geometric horizon . This work was supported financially by NSERC (AAC and ES). JMP would liketo thank AAC for supervising his masters thesis and ES for numerical assistanceand useful discussions, and the Perimeter Institute for Theoretical Physics forhospitality during this work.
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