Geometric horizons in binary black hole mergers

A foliation invariant quasi-local and more geometrical approach, which is an alternative approach to MOTS, has been proposed to discuss BH spacetimes [13]. Since we are primarily interested in numerical applications in 4D to study the (asymmetric) collapse or merger of real BHs, the relevent spacetimes are of general algebraic type away from the horizon. The geometric horizon (GH) conjecture states that [13]: if a BH spacetime is zeroth-order algebraically general, then on the GH the spacetime is algebraically special. Hence a GH can be identified by the alignment type II or D (II/D) discriminant conditions in terms of scalar (curvature) polynomial invariants (SPIs) [15]; that is, a particular set of SPIs vanish on the GH, thereby defining the GH [13].

We note that SPIs may not specify the GH completely in the sense that they may also vanish at fixed points of any isometries and along any axes of symmetry. Unlike AHs, we (again) remark that a GH does not depend on a chosen foliation in the spacetime [11]. In physical problems with dynamical evolution the horizon might not be unique, or may not exist at all, and amendments to the conjecture may be necessary (e.g., it may be appropriate to replace the vanishing of invariants in the definition of a GH as an algebraically special hypersurface, with the conditions that the magnitudes of certain SPIs take their smallest values).

In this letter we wish to evaluate this conjecture by studying the GH numerically in two physically relevant situations: (i) the head-on collision of unequal mass BHs, (ii) two merging, equal mass and non-spinning BHs, during a 4D binary BH (BBH) coalescence [16]. In particular, we shall locally determine the foliation invariant GH based on the algebraic (or Petrov) classification of the Weyl tensor [13]. On the GH, the Weyl tensor is algebraically special. We can use discriminants to determine the necessary conditions for this in terms of simple SPIs, utilizing the algebraic classification (using the boost weight decomposition [15]) of the Weyl and Ricci tensor when treated as curvature operators [17], for the spacetime to be of special algebraic type II or D. The necessary real conditions for the Weyl tensor to be of type II/D are given in [18]. These 2 real conditions are equivalent to the vanishing of the real and imaginary parts of the complex invariant $\mathcal{D}\equiv {I}^{3}-27{J}^{2}=0$ in terms of the complex Weyl spin tensor in the Newman–Penrose (NP) formalism [16]. Alternatively we can use the discriminant analysis to provide the type II/D syzygies expressed in terms of SPIs by treating the Weyl tensor as a trace-free operator acting on the six-dimensional vector space of bivectors [18]. More practical necessary conditions can be obtained by considering the eigenvalue structure of the trace-free symmetric part of the operator C_abcd C^ebcd . This can also be applied to study where the covariant derivative of the Weyl tensor (∇W) is of algebraically special type II/D.

The two applications are complementary. In the head-on collision, the complex part of $\mathcal{D}$ vanishes, so that $\mathcal{D}\equiv {\mathcal{D}}_{\mathrm{r}}$, where ${\mathcal{D}}_{\mathrm{r}}=\text{Re}(\mathcal{D})$, is not of a single sign and hence can be used to locate zeros. Unfortunately the numerics in this case is not always sufficient for conclusive results. In the second application ${\mathcal{D}}_{i}\ne 0$. The complex invariant $\mathcal{D}$ vanishes if and only if its magnitude, $\vert \mathcal{D}\vert =0$. However, numerically finding the level-0 sets of $\vert \mathcal{D}\vert$ precisely is extremely difficult because $\vert \mathcal{D}\vert$ is positive definite. In practice, due to numerical resolution and numerical errors, instead of analyzing the level-0 sets of $\mathcal{D}$, we shall consider the level-ɛ sets for small ɛ (e.g., $\varepsilon \in \left\{3{\times}1{0}^{-4},\enspace 5{\times}1{0}^{-4},\enspace 1{\times}1{0}^{-3}\right\}$).

We wish to study the behaviour of the complex $\mathcal{D}$ (and particularly $\vert \mathcal{D}\vert$), through a BBH merger. Since the Kerr geometry is type D everywhere, it follows that $\mathcal{D}=0$ everywhere for a Kerr BH. It is known that in a BBH merger the merged BHs at late times settle down to a solution well described by the Kerr metric [13]. Thus, for a merger of 2 initially Kerr BHs, a plot of the real part and imaginary part of $\mathcal{D}$ should be roughly zero everywhere at early and late times (and simply give rise to random (numerical) noise); in particular, the numerical noise (artificial zeros) make early time plots somewhat unreliable. However, in the intermediate 'dynamical' region (during the actual merger and coalescence at intermediate times) the Petrov type is I, and the zeros of $\mathcal{D}$ will give us important information.

The horizon of the exact Kerr solution is, in fact, located by the condition that ∇W is of type II/D there (characterized by the fact that various SPIs of ∇W are zero) [19]. The second part of the GH conjectures states that if a BH spacetime is algebraically special (so that on the GH the BH spacetime is automatically algebraically special) and if, for example, ∇W is algebraically general, then on the GH, ∇W is algebraically special [13]. Therefore, it might be useful to also study the algebraic type of ∇W, particularly at early and late times. This might be best done via the NP formalism and the use of appropriate Cartan invariants.

Indeed, in addition to algebraic and differential SPIs, Cartan scalar invariants can also be defined and used [19]. More specifically, for a fixed set of frame vectors a Cartan invariant is a scalar that is constructed from the Weyl or Riemann tensor or any of its covariant derivatives by contracting the Weyl tensor (resp., any of its covariant derivatives) with the frame vectors. Cartan invariants are easier to compute in general than SPIs (because, for example, they are linear in the Weyl tensor and its covariant derivatives). It has been found that in a 'preferred frame' there is evidence that the NP spin coefficients ρ, μ, which are related to Θ_(l), and Θ_(n), can be identified with the surfaces on which ∇W is algebraically special; that is, the GH surfaces are characterized by the conditions that the spin coefficients ρ, μ are zero [13, 20, 21]. Indeed, the spin coefficients ρ and μ were utilized in the numerical studies of [13, 20]. In future we could further investigate ∇W or the spin coefficients ρ, μ (in the preferred frame) in the current applications, but this will present some numerical and analytic difficulties. In addition, the question of whether these spin coefficients take on a special meaning in a 'preferred frame' will be further investigated elsewhere.

1.1.1. Examples and motivation

1.1.2. Numerics

We utilize the numerical data from [7], based on the Einstein toolkit and enforcing axisymmetry and starting with Brill–Lindquist initial data representing a BBH system at a moment of time-symmetry, for the head-on axisymmetric collision of two non-spinning unequal mass BHs with mass ratio m(z > 0)/m(z < 0) = (0.8)/(0.5). In figure 1 we present the time plots for t = 0.5–5.5 (and at later times t = 8, 9); by t = 1/2 the outer MOTS has already formed. We lose numerical resolution at later times when the horizons get too close to the punctures. Therefore, we did not look for MOTs there.

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Since in this simulation ${\mathcal{D}}_{i}=\text{Im}(\mathcal{D})=0$ (i.e., ${\mathcal{D}}_{\mathrm{r}}\equiv \text{Re}(\mathcal{D})=\mathcal{D}$), to study the level-0 sets of $\mathcal{D}$ it suffices to study the level-0 sets of ${\mathcal{D}}_{\mathrm{r}}$. In particular, the changing of sign through a zero of D = D_r reveals a MOTS (and there are no problems regarding resolution since we are not considering a positive definite quantity as in second application). In figure 1, we compare the interpolating sequence of MOTSs with the level-0 sets of ${\mathcal{D}}_{\mathrm{r}}$. We plot ${\mathcal{D}}_{\mathrm{r}}$ vs x and z for y = 0. We obtain the data for y ≠ 0 by using cylindrical coordinates (x, z, θ) to describe the system, with θ describing rotations around the z axis in the right-hand sense, so that the data for θ = θ₀ is obtained by rotating the data for θ = 0 through the angle θ₀.

It should be noted that ${\mathcal{D}}_{\mathrm{r}}=0$ at many points on the z axis, but these points have been omitted in figure 1. Indeed, in [13] it was noted that $\mathcal{D}$ could vanish additionally on axes of symmetry, which seems to be the case in this present simulation, as z = 0 is an axis of symmetry for the simulation, as described in the previous paragraph. We also notice that ${\mathcal{D}}_{\mathrm{r}}=\mathcal{D}=0$ in a dense region in the interior of ${\mathcal{S}}_{1}$ and ${\mathcal{S}}_{2}$. As noted above, at early times the configuration is close to 2 individual Kerr BHs so that the spacetime is of type D almost everywhere and so numerical noise (artificial zeros) make early time plots somewhat unreliable with no clear closed curves visible.

Aside from this region, the level-0 set of ${\mathcal{D}}_{\mathrm{r}}$ is seen to be partitioned into 2 simple closed curves at time t = 3. One such curve interpolates between ${\mathcal{S}}_{i}$ and ${\mathcal{S}}_{c}$, while the other curve encircles S_c at time t = 3. At t = 3.5, a third simple closed curve becomes visible, which lies in the interior of ${\mathcal{S}}_{2}$ (purple), lying in the component of ${\mathcal{S}}_{i}$ where z ⩽ 0. Note that the purple curves ${\mathcal{S}}_{1}$ and ${\mathcal{S}}_{2}$ happen to be overlaid by the blue ${\mathcal{S}}_{i}$. Then at times t = 4, 4.5, 5, the two visible closed curves shrink so that they reasonably approximate ${\mathcal{S}}_{i}$ and ${\mathcal{S}}_{2}$ (overlaid by the bottom component of ${\mathcal{S}}_{i}$). (The top component of ${\mathcal{S}}_{i}$, which coincides with ${\mathcal{S}}_{1}$ (above z = 0), is tracked by a dense region of the ${\mathcal{D}}_{\mathrm{r}}$ level curve). However, one of these two visible simple closed curves no longer surround ${\mathcal{S}}_{i}$ (and ${\mathcal{S}}_{1}$ and ${\mathcal{S}}_{2}$); instead, it shrinks in average diameter. At later times the third simple closed curve is not visible in the frame; it is located further out compared to the scale used. The bottom left and right panels of figure 1 indicate the behaviour of the level-0 sets of ${\mathcal{D}}_{\mathrm{r}}$ at times t = 8.0 and t = 9.0, which show a zoomed out view of the outer horizon. For example, t = 9.0 shows part of the spherical GH that encloses both BHs. These plots show that the level-0 sets of ${\mathcal{D}}_{\mathrm{r}}$ could define a unique GH. However, further investigation is needed, especially at later times when the outer MOTS (the AH) is close to isolated again and has reached Kerr. This might require going up to at least t = 20 (where the horizon mass becomes constant), at which time a GH might be present near the (outer) AH. At later times, however, when the geometry is nearly Kerr and the spacetime is type D everywhere, we may also need to study the covariant derivative of the Weyl tensor.

We study the algebraic properties of the Weyl tensor through a quasi-circular merger of two non-spinning, equal mass BHs. The GH is theoretically identified by the level-0 set of the complex invariant $\mathcal{D}$. We find strong evidence that the level-ɛ sets of $\vert \mathcal{D}\vert$ track a unique smooth GH through all stages of the BBH merger, by analyzing the time evolution of various level-ɛ sets of $\vert \mathcal{D}\vert$ where, in particular, ɛ_i ≡ 3 × 10⁻⁴, 5 × 10⁻⁴, 1 × 10⁻³ for (i = 1–3), respectively. We note that many additional figures (including those describing the inner and outer MOTS) are presented in [38].

The figures provide plots of various level sets of $\vert \mathcal{D}\vert$ as functions of $(x,y)\in {\mathbb{R}}^{2}$ at a fixed spatial coordinate value of z = 0.3125 and at selected instances of the time parameter, t, where t = 0 indicates the start of the numerical computation. In each of the figures, the data corresponding to x < 0 was obtained by rotating the data corresponding to x > 0 by 180 degrees about the x = y = 0 axis. In [38], the contour plots of $\vert \mathcal{D}\vert$ (and ${\mathcal{D}}_{\mathrm{r}}$ and other complex quantities) and its level-0 sets and various other level-ɛ sets at all times t = 0–42 and in more detail and with magnified resolution were presented. Here we shall illustrate the qualitative features of the contour plots of the level-ɛ sets $\vert \mathcal{D}\vert$ on a log scale for t = 12, 16, 20. The overlaid green, red and white contours in each first panel (top left-hand side) of the figures are the level-ɛ₁, level-ɛ₂ and level-ɛ₃ sets of $\vert \mathcal{D}\vert$, respectively. The blue dots in the figures give the centroids of the MOTSs of the two initial BHs as they evolve and serve to track the positions of these BHs through the merging process. At early times, when the configuration is close to 2 Kerr BHs and the spacetime is approximately of type D, the time plots are somewhat unreliable.

At early times (e.g., at t = 8), each of the level-ɛ sets are partitioned into pairs of simple disjoint closed curves, each of which contains the centroid of the MOTS of each of the two initial BHs. At intermediate times (e.g., t = 16), the red level-ɛ₂ set and the white level-ɛ₃ set of $\vert \mathcal{D}\vert$ each form a third simple closed curve between the centroids of the MOTSs of the two initial BHs, which is centred at the origin (this occurs a little earlier for the green level-ɛ₁ set [38]). At later times (e.g., at t = 20), for each respective ɛ_i , the multiple simple closed curves partitioning the level-ɛ set of $\vert \mathcal{D}\vert$ have joined so that each level-ɛ curve is now a single simple closed curve surrounding the 2 BHs. It follows that the level-ɛ_i curves at each t form an invariantly defined, foliation invariant horizon that contains each separate BH at early times, and contains the merged BH at late times. In addition, we observe that the level-ɛ_i contours are very close to each other, showing that the level-ɛ sets vary continuously with ɛ. We also observe that if ɛ_I ⩽ ɛ_J , then the 2D area enclosed by the level-ɛ_I curve encloses the 2D area enclosed by the level-ɛ_J curve, indicating that $\vert \mathcal{D}\vert$ decreases on average with average distance from the centroids of the MOTSs.

We compare the white level-1 × 10⁻³ contours of $\vert \mathcal{D}\vert$ with points on the surface of the MOTSs of the 2 initial BHs at times t = 12, 16, 20 in the second panel (top right-hand side) of figures 2–4. The blue curves in the figures indicate the (x, y) coordinates of a z = 0.031 25 slice of the points on the 'spherically averaged' MOTSs of each initial BH (i.e., the 2D sphere centred at the centroid of the MOTS and whose radius is the average radius of the MOTS). The centroid of this MOTS is again given as the central blue dot. The light sky blue curves mark the (x, y) coordinates along each 'actual' MOTS of each initial BH (i.e., without the spherical approximation). We note that the MOTSs of the two initial BHs are nearly spherically symmetric surfaces and that the white level-1 × 10⁻³ set of $\vert \mathcal{D}\vert$ coincides closely with the MOTSs (both spherically averaged and exact), especially at early times. This lends support to the choice of the white level-1 × 10⁻³ sets of $\vert \mathcal{D}\vert$ as a representative approximation to the level-0 set of $\vert \mathcal{D}\vert$. In figure 4 for t = 20 there is also an outermost MOTS (the AH) which is displayed in the final panel on a different scale.

The evolution of the level-ɛ curves through the quasi-circular BBH merger in the figures is qualitatively similar to the sequence of MOTS that take place during the head-on collision simulation in [7] (see earlier) and the GH in the KT analysis [23]. However, our numerical simulations were not run to sufficiently late times to study the details of the bifurcation.

The complex invariant $\mathcal{D}$ vanishes if and only if its magnitude, $\vert \mathcal{D}\vert =0$. However, numerically finding the level-0 sets of $\vert \mathcal{D}\vert$ precisely is extremely difficult because $\vert \mathcal{D}\vert$ is a positive definite quantity, so that any numerical errors would provide a positive contribution. Furthermore, in the numerical simulations it is possible that the actual zeros of $\vert \mathcal{D}\vert$ do not occur at any points which are sampled for the discrete mesh being used. Thus, the level-ɛ sets of $\vert \mathcal{D}\vert$ could indeed approximate the level-0 sets $\mathcal{D}$ for an appropriate preferred ɛ-value.

If the value of $\vert \mathcal{D}\vert$ itself is small, then the locations of the local minima of $\vert \mathcal{D}\vert$ could possibly indicate the positions of the actual zeros, which would be missed due to the numerical issues described above. It is conceivable that the GH conjecture should be modified so that the GH is defined as the set of points where $\vert \mathcal{D}\vert$ reaches the local minimum instead of being identically zero. If this were the case, then locating the local minima of $\vert \mathcal{D}\vert$ would locate the GH precisely instead of approximating it. However, further evidence from the analysis of ${\mathcal{D}}_{\mathrm{r}}$, as described later, perhaps suggests that this is not the case.

Thus, to further study the zeros of $\vert \mathcal{D}\vert$, which would indicate the zeros of $\mathcal{D}$, we study the positions of the local minima of $\vert \mathcal{D}\vert$. To find these positions, it is sufficient to track the positions of the local minima of $\vert \mathcal{D}\vert$ along so-called 'slice plots' of $\vert \mathcal{D}\vert$ vs y for a fixed x. [Along each slice plot, we find the values of y = y_min, where $\vert \mathcal{D}\vert$ assumes a local minimum value, and show that the point (x_min, y_min) accurately represents the local minimum, and the corresponding points are then recorded. We restrict our attention to finding the positions of the local minima of $\vert \mathcal{D}\vert$ whose corresponding $\vert \mathcal{D}\vert$ values lie within the range $\left[1{\times}1{0}^{-4},\enspace 1.2{\times}1{0}^{-3}\right]$ which contains the set of values of ɛ_i being considered for the level-ɛ sets of $\vert \mathcal{D}\vert$.] Having done this for all possible fixed x, we then present plots of the locations of these local minima of $\vert \mathcal{D}\vert$ with green dots at selected times, t = 16 and 20, in panels 3 of figures 3 and 4. These plots demonstrate that at time t = 16 (and earlier), the positions of the local minima of $\vert \mathcal{D}\vert$ appear to track the green level-ɛ₁ sets, while at time t = 20 (and later), these local minima appear to track more closely the white level-ɛ₃ sets. Therefore, the figures show that the positions of the local minima of $\vert \mathcal{D}\vert$ accurately track the zeros of $\vert \mathcal{D}\vert$, and hence the GH, during the BBH merger. We will now argue that all local minima of $\vert \mathcal{D}\vert$ track closely the level-0 sets of ${\mathcal{D}}_{\mathrm{r}}$.

The problem of estimating the level-0 sets of the complex $\mathcal{D}$ cannot be definitively resolved by analyzing $\vert \mathcal{D}\vert$ because, as previously noted, $\vert \mathcal{D}\vert$ is a positive definite quantity and the discrete resolution imposed by the numerical simulation does not necessarily allow one to accurately find level-0 sets of $\vert \mathcal{D}\vert$. Thus, to gain further insight into the GH through the BBH merger, it is therefore helpful to analyze quantities which change sign through a zero. This will qualitatively demonstrate the existence of level-0 sets. This avoids some of the problems of numerical resolution that occur when using the local minima of $\vert \mathcal{D}\vert$ to estimate the zeros of $\vert \mathcal{D}\vert$ (and hence $\mathcal{D}$).

In panels 4 of figures 3 and 4 we compare the level-−0.01 sets (in yellow) and the level-+0.01 sets (in lime green) of ${\mathcal{D}}_{\mathrm{r}}$ at times t = 16, 20 with the white level-1 × 10⁻³ sets of $\vert \mathcal{D}\vert$. [In these figures, the grey regions in each of the frames correspond to regions where $-0.01{< }{\mathcal{D}}_{\mathrm{r}}{< }0.01$, as indicated in the colourbar, but the black regions correspond to regions where ${\mathcal{D}}_{\mathrm{r}}{\geqslant}1$ and the white regions correspond to regions where ${\mathcal{D}}_{\mathrm{r}}{\leqslant}-1$]. We know that a positive level set and a nearby negative level set indicates a sign change of the quantity being plotted. Hence, there must be a surface bounded by the level ±0.01 sets of ${\mathcal{D}}_{\mathrm{r}}$ across which ${\mathcal{D}}_{\mathrm{r}}$ changes sign. This surface gives the level-0 set of ${\mathcal{D}}_{\mathrm{r}}$. By estimating the zeros of ${\mathcal{D}}_{\mathrm{r}}$ in this manner, we reduce the possibility of numerical noise that comes with plotting the level-0 sets of ${\mathcal{D}}_{\mathrm{r}}$ directly. Upon inspection of each figure, we see that the level-±0.01 sets of ${\mathcal{D}}_{\mathrm{r}}$ are densely clustered around the centroids of the MOTSs of the initial BHs and occur in close proximity with but are contained in the interior of the level-1 × 10⁻³ sets of $\vert \mathcal{D}\vert$. It follows that there is strong evidence that the level-1 × 10⁻³ set of $\vert \mathcal{D}\vert$ well approximates the elusive level-0 sets of the complex invariant $\mathcal{D}$.

We utilize the notation described earlier (in section 2). At around t = 18.5–18.75 the third MOTS ${\mathcal{S}}_{i}$ first appears and bifurcates to give rise to the fourth MOTS ${\mathcal{S}}_{c}$, and after t = 19 these appear in the figures. The outermost MOTS ${\mathcal{S}}_{c}$ (the AH) can be seen in the numerics but is outside of the domain in the current figures. Indeed, the outermost MOTS at late times is so big that the entire two dots and S₁ and S₂ and the inner S_i are all squashed in the origin. In figure 4 (panel 2) for t = 20, the 'inner' MOTS ${\mathcal{S}}_{i}$ is displayed as purple dots. The actual MOTS ${\mathcal{S}}_{i}$ as plotted is calculated exactly, and is traced very well by the spherically averaged MOTS (shown by the outer light pink oval figure, which overlays the purple dots but is not clearly visible). The outer MOTS ${\mathcal{S}}_{c}$ is present, but is out of view and cannot be seen in the figure displayed.