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Geometric Horizons ∗ Alan A Physics Letters B 771 (2017) 131–135 Contents lists available at ScienceDirect Physics Letters B www.elsevier.com/locate/physletb Geometric horizons ∗ Alan A. Coley a, David D. McNutt b, , Andrey A. Shoom c a Department of Mathematics and Statistics, Dalhousie University, Halifax, Nova Scotia, B3H 3J5, Canada b Faculty of Science and Technology, University of Stavanger, N-4036 Stavanger, Norway c Department of Mathematics and Statistics, Memorial University, St. John’s, Newfoundland and Labrador, A1C 5S7, Canada a r t i c l e i n f o a b s t r a c t Article history: We discuss black hole spacetimes with a geometrically defined quasi-local horizon on which the Received 21 February 2017 curvature tensor is algebraically special relative to the alignment classification. Based on many examples Accepted 2 May 2017 and analytical results, we conjecture that a spacetime horizon is always more algebraically special (in Available online 19 May 2017 all of the orders of specialization) than other regions of spacetime. Using recent results in invariant Editor: M. Trodden theory, such geometric black hole horizons can be identified by the alignment type II or D discriminant conditions in terms of scalar curvature invariants, which are not dependent on spacetime foliations. The above conjecture is, in fact, a suite of conjectures (isolated vs dynamical horizon; four vs higher dimensions; zeroth order invariants vs higher order differential invariants). However, we are particularly interested in applications in four dimensions and especially the location of a black hole in numerical computations. © 2017 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. 1. Introduction directed null normal to the foliation vanishes, while the expansion of the other future directed null normal is negative (a FOTH has Black holes, which are exact solutions in general relativity (GR) the additional condition that the directional derivative along the (representing, for example, physical objects formed out of the grav- second null direction is negative). itational collapse of fuel-exhausted stars), are characterized by the A dynamical horizon is particularly well-suited to analyze dy- boundary of the region from where light can still travel to asymp- namical processes involving black holes, such as black hole growth totic null infinity, called the event horizon, which is usually iden- and coalescence. The area of a dynamical horizon necessarily in- tified as the surface of the black hole and relates its area to the creases with time [1]. An explicit example of a dynamical horizon entropy. The event horizon is essentially a global (teleological) ob- is given by the Vaidya spacetime which admits spherically sym- ject, since it depends on the entire future history of the spacetime metric MTSs [3,4]. For a given mass function, the Vaidya spacetime [1]. also provides explicit examples of the transition from the dynami- There has been much effort to give a general quasi-local de- cal to isolated horizons. If a hypersurface admits a dynamical hori- scription of a dynamical black hole [1,2]. Of particular interest zon structure, it is unique. However, because a spacetime may have are quasi-local objects called marginally trapped tubes (MTTs) or several distinct black holes, it may admit several distinct dynam- trapping horizons, and the special cases of dynamical horizons or ical horizons. For dynamical horizons which are also FOTHs, two future outer trapping horizons (FOTHs); in numerical work, these non-intersecting horizons generally either coincide or one is con- are also called apparent horizons. MTTs are hypersurfaces foliated tained in the other [1]. by (closed compact space-like two-dimensional (2D) submanifolds It is believed that closed MTSs constitute an important ingre- without boundary) marginally trapped surfaces (MTSs) in which the dient in the formation of black holes, which motivates the idea of expansion of one of the null normals vanishes and the other is using MTTs as viable replacements for the event horizon of black non-positive. A dynamical horizon is a smooth 3D submanifold of holes [2]. Unfortunately, since the 2D apparent horizons depend spacetime foliated by MTSs such that the expansion of one future- on the choice of a reference foliation of spacelike hypersurfaces, MTTs and consequently trapping horizons and dynamical horizons are highly non-unique [5]. There have been some attempts to pro- vide a physically sound criterion for selecting a preferred MTT such * Corresponding author. E-mail addresses: [email protected] (A.A. Coley), [email protected] as, for example, in which the shear scalars along ingoing/outgoing (D.D. McNutt), [email protected] (A.A. Shoom). null directions foliated by 2D spacelike surfaces vanish [6]. http://dx.doi.org/10.1016/j.physletb.2017.05.004 0370-2693/© 2017 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. 132 A.A. Coley et al. / Physics Letters B 771 (2017) 131–135 W ≡− 3 + − = Realistic black holes interact with their environment and are 1 11W 2 33W 2 W 4 18W 6 0, (3) consequently dynamical. The gravitational collapse leading to black W ≡ (W 2 − 2W )(W 2 + W )2 hole formation is also a highly dynamical process. It is crucial to 2 2 4 2 4 + 2 − 2 − + 3 = locate a black hole locally, which may not rely on the existence of 18W 3 (6W 6 2W 3 9W 2 W 4 3W 2 ) 0, (4) an event horizon alone. A significant fraction of research in numer- where ical relativity aims at predicting with high precision the waveforms of gravitational waves generated in the merger of compact-object 1 abcd 1 cd pqab W 2 = CabcdC , W 3 = CabcdC C , binary systems or in stellar collapse to form black holes. Compari- 8 16 pq son with templates played a crucial role in the recent observations 1 = cd pq rsab W 4 CabcdC C rsC , of gravitational waves from black hole mergers by the LIGO Collab- 32 pq oration [7]. 1 = cd pq rs tu vwab In numerical studies of time-dependent collapse, it is often W 6 CabcdC C rsC C C . (5) 128 pq tu vw more practical to track apparent horizons or trapping horizons [8]. In contrast with the event horizon, which is a global concept de- These 2 real conditions are equivalent to the real and imaginary 3 − 2 = fined using the global structure of spacetime, the apparent horizon parts of the complex syzygy I 27 J 0in terms of the complex is a quasi-local concept and is intrinsically foliation-dependent. In Weyl tensor in the Newman–Penrose (NP) formalism [14]. this paper we propose a foliation invariant and more geometrical Alternatively we can use the discriminant analysis to provide ∈[ ] approach, which is possible due to recent results in invariant the- the type II/D syzgies expressed in terms of the SPIs W i (i 1, 6 , ory. defined above) by treating the Weyl tensor as a trace-free oper- ator acting on the 6-dimensional vector space of bivectors [12] (however, these conditions are very large). More practical nec- 2. Scalar polynomial curvature invariants essary conditions can be obtained by considering the trace-free ebcd e symmetric operator CabcdC −2W 2δ ; applying the discriminant The algebraic classification of the Weyl tensor and the Ricci a analysis we find the coefficients of the characteristic equation are tensor in arbitrary dimensions using the boost weight decompo- = 2 − w2 8(W 2 2W 4) (and similarly for w3, w4) and so the neces- sition [9] can be refined utilizing the restricted eigenvector and sary condition for this operator to be type II/D is given by equation eigenvalue structure of their associated curvature operators [10], (1) (with the si replaced by wi ). allowing for necessary conditions to be defined for a particular The alignment classification can be applied to any rank tensor. algebraic type in terms of a set of discriminants. A scalar poly- To consider whether the covariant derivatives of the Ricci tensor, nomial curvature invariant of order k (or, in short, a scalar polyno- Rab;cd..., are of type II or D, we can use the eigenvalue structure mial invariant or SPI) is a scalar obtained by contraction from of the operators associated with the derivatives of the Ricci cur- a polynomial in the Riemann tensor and its covariant derivatives vature and impose the type II/D necessary conditions. This can be up to the order k. Black hole spacetimes are completely character- repeated for the Weyl tensor and in arbitrary dimensions [15]. For ized by their SPIs [11]. In particular, we can use discriminants to example, for the covariant derivative of the Weyl tensor, Cabcd;e , study the necessary conditions in arbitrary dimensions, in terms in 4D we can consider the second order symmetric and trace-free of simple SPIs, allowing for the algebraic classification of the 1 a operator T b defined by: higher dimensional Weyl and Ricci tensor when treated as curva- ture operators, for the spacetime to be of algebraic type II or D 1 a cdef ;a 1 a 1 T ≡ C Ccdef ;b − δ I2 (6) [12]. b 4 b For example, in 5D the necessary condition for the trace-free 1 ≡ abcd;e 1 where I2 C Cabcd;e , and we have the corresponding 4th, Ricci tensor, Sab = Rab − Rgab, to be of algebraic type II/D is that 1 1 1 5 6th, 8th order invariants I4, I6, I8. Computing the coefficients 5 the discriminant (SPI) D is zero, and the necessary conditions 1 1 1 1 1 2 S 5 of the characteristic equation we obtain s2 =− I4 + I (and 10 2 8 2 for the Weyl tensor to be of type II/D is that the SPIs D (i = 1 1 W i similarly for s3, s4).
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