Introduction to dynamical horizons in numerical relativity

This paper presents a quasilocal method of studying the physics of dynamical black holes in numerical simulations. This is done within the dynamical horizon framework, which extends the earlier work on isolated horizons to time-dependent situations. In particular: (i) We locate various kinds of marginal surfaces and study their time evolution. An important ingredient is the calculation of the signature of the horizon, which can be either spacelike, timelike, or null. (ii) We generalize the calculation of the black hole mass and angular momentum, which were previously defined for axisymmetric isolated horizons to dynamical situations. (iii) We calculate the source multipole moments of the black hole which can be used to verify that the black hole settles down to a Kerr solution. (iv) We also study the fluxes of energy crossing the horizon, which describes how a black hole grows as it accretes matter and/or radiation. We describe our numerical implementation of these concepts and apply them to three specific test cases, namely, the axisymmetric head-on collision of two black holes, the axisymmetric collapse of a neutron star, and a nonaxisymmetric black hole collision with nonzero initial orbital angular momentum.

Figure 1

A dynamical horizon $H$ bounded by MOTSs $S_1$ and $S_2$. ${\ell }^a$ is the outgoing null normal, $n^a$ is the ingoing null normal, ${\widehat{r}}^a$ is the unit spacelike normal to the cross sections, and ${\widehat{\tau }}^a$ is the unit timelike normal to $H$. $\Sigma$ is a Cauchy surface intersecting $H$ in a 2-sphere $S$. $T^a$ is the unit timelike normal to $\Sigma$ and $R^a$ is the unit spacelike outward pointing vector normal to $S$ and tangent to $\Sigma$.

Figure 2

Two MOTSs $S_{(1)}$ and $S_{(2)}$ surrounded by a common MOTS $S_{\mathrm{out}}$. Spheres lying just inside these FMOTSs must have negative outgoing expansion. Thus, there must be a inner trapped horizon $S_{\mathrm{in}}$ inside $S_{\mathrm{out}}$ which encloses $S_{(1)}$ and $S_{(2)}$.

Figure 3

Coordinate shapes of the horizons at $t=1$ in the $xz$ plane. A common horizon has formed, and the inner and outer common horizons have already separated. Compare Fig. 2.

Figure 4

Determinant of the horizon world tube’s three-metric vs latitude $\theta$ at $t=0.6$ and $t=1$. The individual MTTs are null, i.e., $\det \tilde{q}=0$ (up to numerical errors). The common outer MTT is spacelike (i.e., $\det \tilde{q}>0$) and it tends to null at late times. The inner common MTT is partially timelike at $t=0.6$; later it becomes completely timelike.

Figure 5

Irreducible mass vs time for the individual and the common MTTs. The outer common MTT grows and accretes mass, while the inner MTT shrinks and loses mass.

Figure 6

Energy flux and shear $|{\sigma }_{(\overline{\ell })}{|}^2$ through the outer common horizon vs latitude $\theta$ at $t=0.6$, and the total energy flux vs time. The shear vanishes at the poles and the black hole settles down exponentially.

Figure 7

Some mass multipole moments vs time for the inner and outer MTTs for the head-on collision. The multipole moments for the outer horizon all approach their Schwarzschild values (i.e., 0) but the inner horizon does not seem to do so.

Figure 8

Coordinate shapes of the MOTSs at $t=18$ for the nonaxisymmetric black hole collision. Note that the individual horizons are locked in place through the corotating coordinate system and through an adaptive shift condition.

Figure 9

Determinant of the MTT three-metric at $t=18$. As in the head-on case, the outer MTT is purely spacelike while the inner MTT is partly spacelike and partly timelike. At later times, it becomes purely timelike. The individual MTTs are null at this time.

Figure 10

A plot of the irreducible mass $M_{\mathrm{irr}}=\sqrt{A/16\pi }$ as a function of time for the outer an inner MTTs in the nonaxisymmetric black hole collision. As expected, the outer MTT has increasing area while the inner MTT shrinks.

Figure 11

Energy flux through the common horizon and shear $|{\sigma }_{\overline{(\ell )}}{|}^2$ on the horizon at $t=18.4$ in $(\theta ,\varphi )$-coordinates. The spacetime is not axisymmetric, and since it contained two inspiralling black holes, there is no mirror symmetry across the $x-z$ or $y-z$ planes either. The energy flux shows this asymmetry clearly.

Figure 12

Lie derivative of the two-metric ${\mathcal{L}}_{\phi }{\tilde{q}}_{ab}$ at $t=18$ on the $\varphi =0$ line. The two-metric ${\tilde{q}}_{ab}$ is also shown for comparison. The quantity shown in the plots are actually the norms $\sqrt{\sum _{ab}({\mathcal{L}}_{\phi }{\tilde{q}}_{ab}{)}^2}$ and $\sqrt{\sum _{ab}({\tilde{q}}_{ab}{)}^2}$ in the coordinate system $(\theta ,\varphi )$ on the horizon. The vector field $\phi$ is Killing on the equator (see main text), but not everywhere. This shows that the horizon is not (yet) axisymmetric. We expect it to become axisymmetric at later times. Note that we have only shown the plots along the $\varphi =0$ curve and we do not have axisymmetry here.

Figure 13

The average coordinate radius and the area radius as a function of time for the outer and inner MTTs for the neutron star collapse. The inner horizon is not to be trusted after $t\approx 140$ due to lack of resolution, since its coordinate radius has become very small by that time.

Figure 14

Average of the determinant of the horizon world tube’s three-metric vs time, and vs latitude $\theta$ at $t=138.24$ for the inner and outer horizons for the neutron star collapse.

Figure 15

The total mass $M_H$, and angular momentum $J$ as a function of time for the outer and inner MTTs for the neutron star collapse.

Figure 16

Horizon mass quadrupole moment $M_2$ and angular momentum octopole moment $J_3$ vs time for the neutron star collapse. For comparison, the values for a Kerr black hole with the same ADM mass and angular momentum as the initial data are also shown.