PHYSICAL REVIEW D 83, 104018 (2011) Horizon dynamics of distorted rotating black holes Tony Chu,1 Harald P. Pfeiffer,2 and Michael I. Cohen1 1Theoretical Astrophysics 350-17, California Institute of Technology, Pasadena, California 91125, USA 2Canadian Institute for Theoretical Astrophysics, 60 St. George Street, University of Toronto, Toronto, ON M5S 3H8, Canada (Received 11 November 2010; published 6 May 2011) We present numerical simulations of a rotating black hole distorted by a pulse of ingoing gravitational radiation. For strong pulses, we find up to five concentric marginally outer trapped surfaces. These trapped surfaces appear and disappear in pairs, so that the total number of such surfaces at any given time is odd. The world tubes traced out by the marginally outer trapped surfaces are found to be spacelike during the highly dynamical regime, approaching a null hypersurface at early and late times. We analyze the structure of these marginally trapped tubes in the context of the dynamical horizon formalism, computing the expansion of outgoing and incoming null geodesics, as well as evaluating the dynamical horizon flux law and the angular momentum flux law. Finally, we compute the event horizon. The event horizon is well-behaved and approaches the apparent horizon before and after the highly dynamical regime. No new generators enter the event horizon during the simulation. DOI: 10.1103/PhysRevD.83.104018 PACS numbers: 04.25.dg, 04.25.DÀ Quasilocal horizons can be computed locally in time, I. INTRODUCTION and so are used instead to locate a black hole during the In the efforts by the numerical relativity community evolution. Of particular interest is a marginally outer leading up to the successful simulation of the inspiral trapped surface (MOTS), which is a spatial surface on and merger of two black holes, analyses of single black which the expansion of its outgoing null normal vanishes holes distorted by gravitational radiation have offered a [6]. The use of MOTSs is motivated by several results. convenient and simpler setting to understand the non- When certain positive energy conditions are satisfied, an linear dynamics during the late stages of binary black MOTS is either inside of or coincides with an event horizon hole coalescence. For this purpose, initial data for a [6,7]. The presence of an MOTS also implies the existence Schwarzschild black hole plus a Brill wave was presented of a spacetime singularity [8]. In an evolution, the MOTSs in [1], which was both time symmetric and axisymmetric. located at successive times foliate a world tube, called a In highly distorted cases, the apparent horizon could marginally trapped tube (MTT). MTTs have been studied develop very long, spindlelike geometries. If the event in the context of trapping horizons [9,10], isolated horizons horizon can show similar behavior, this would raise in- [11–13], and dynamical horizons [14–16]. triguing questions related to the hoop conjecture [2]. The Both the event horizon and an MTT react to infalling work of [1] was extended to distorted rotating black holes matter and radiation, although their behaviors can be quite in [3], where the apparent horizon served as a useful tool different in highly dynamical situations. Being a null sur- to examine the quasinormal oscillations of the black hole face, the evolution of the event horizon is governed by the geometry as it relaxed in an evolution. Further studies null Raychaudhuri equation [17], so that even though its have extracted the gravitational waves emitted by the area never decreases, in the presence of infalling matter black hole [4], and compared the apparent and event and radiation the rate of growth of its area decreases and horizons [5]. can even become very close to zero [18]. Since an MTT is We continue this line of investigation here, while incor- determined by quasilocal properties of the spacetime, its porating various modern notions of quasilocal horizons reaction to infalling matter and radiation is often much that have emerged in recent years. Our emphasis is on more intuitive. An MTT is usually spacelike (e.g., a horizon properties during the highly dynamical regime, dynamical horizon) in such situations, although further and no symmetries are present in our initial data and scrutiny has revealed that MTTs can exhibit various in- evolutions. The utility of quasilocal horizons can be im- triguing properties of their own. For example, an MTT may mediately appreciated when one wants to perform a nu- become timelike and decrease in area [19], or even have merical evolution of a black hole spacetime. One must be sections that are partially spacelike and partially timelike able to determine the surface of the black hole at each time [20]. In a numerical simulation, such behavior is often in order to track the black hole’s motion and compute its indicated by the appearance of a pair of new MTTs at a properties, such as its mass and angular momentum. given time, accompanied by a discontinuous jump in the However, the event horizon, which is the traditional notion world tube of the apparent horizon, or outermost MOTS. of a black hole surface, can only be found after the entire In this paper, we investigate the behavior of MTTs and future history of the spacetime is known. the event horizon in the context of a rotating black hole 1550-7998=2011=83(10)=104018(17) 104018-1 Ó 2011 American Physical Society TONY CHU, HARALD P. PFEIFFER, AND MICHAEL I. COHEN PHYSICAL REVIEW D 83, 104018 (2011) distorted by an ingoing pulse of gravitational waves. First, and a set of constraint equations, we construct a series of initial data sets in which the R þ K2 À K Kij ¼ 0; (5) amplitude of the gravitational waves varies from small to ij ij ij large, which are then evolved. We focus on the evolution rjðK À g KÞ¼0: (6) with the largest distortion of the black hole, in which the In the above, L is the Lie derivative, ri is the covariant mass of the final black hole is more than double its initial ij derivative compatible with gij, R ¼ g Rij is the trace of value. During the evolution, the world tube of the apparent ¼ ij horizon jumps discontinuously when the gravitational the Ricci tensor Rij of gij, and K g Kij is the trace of Æ waves hit the black hole, and as many as five MTTs are the extrinsic curvature Kij of t. found at the same time. Some of these MTTs decrease in Next, a conformal decomposition of various quantities is ~ area with time, although we find that all the MTTs during introduced. The conformal metric gij and conformal factor the dynamical stages of our evolution are spacelike and c are given by dynamical horizons. Moreover, all these MTTs join to- 4 gij ¼ c g~ij; (7) gether as a single dynamical horizon. Their properties are further analyzed using the dynamical horizon flux law [15], the time derivative of the conformal metric is denoted by which allows one to interpret the growth of the black hole ~ ¼ ~ in terms of separate contributions. We also evaluate the u ij @tgij; (8) angular momentum flux law based on the generalized ij and satisfies u~ijg~ ¼ 0, while the conformal lapse is given Damour-Navier-Stokes equation [21]. Finally, we locate by N~ ¼ c À6N. Equations. (5) and (6), and the trace of (4) the event horizon and contrast its behavior with that of can then be written as the MTTs. 1 1 1 The organization of this paper is as follows. Section II r~ 2 c À c R~ À c 5K2 þ c À7A~ A~ij ¼ 0; (9) details the construction of the initial data sets and Sec. III 8 12 8 ij describes the evolutions. Section IV introduces some defi- 1 1 2 ~ 6 ~ nitions about MOTSs, and the methods used to locate them. r~ ðL Þij À r u~ij À c riK ¼ 0; (10) Section V discusses the MTTs foliated by the MOTSs, the j 2N~ j 2N~ 3 determination of their signatures, and the fluxes of energy 1 5 7 and angular momentum across them. The emphasis is on r~ 2ð ~c 7Þð~c 7Þ ~ þ c 4 2 þ c À8 ~ ~ij the case with the largest distortion of the initial black hole, N N 8 R 12 K 8 AijA as is the remainder of the paper. Section VI explains how 5 ¼c ð@ K À k@ KÞ: (11) we find the event horizon, and contrasts its properties with t k the MTTs. Section VII presents some concluding remarks. ~ In the above, ri is the covariant derivative compatible with Finally, the appendix offers some insight on our results in g~ , R~ ¼ g~ijR~ is the trace of the Ricci tensor R~ of g~ , L~ is light of the Vaidya spacetime. ij ij ij ij the longitudinal operator, 2 ðL~ Þij ¼ r~ i j þ r~ j i À ~ijr~ k II. INITIAL DATA 3 g k ; (12) Initial data sets are constructed following the method of ~ij [22], which is based on the extended conformal thin sand- and A is wich formalism. First, the 3 þ 1 decomposition of the 1 A~ ij ¼ ððL~ Þij À u~ijÞ; (13) spacetime metric is given by [23,24] 2N~ ð4Þ 2 ¼ ds gdx dx ; (1) which is related to Kij by 2 2 i i j j 1 ¼N dt þ gijðdx þ dtÞðdx þ dtÞ; (2) ¼ c À10 ~ þ Kij Aij 3 gijK: (14) where g is the spatial metric of a t ¼ constant hypersur- ij ~ ~ face Æ , N is the lapse function, and i is the shift vector.