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Register a new userCartan Invariants and Event Horizon Detection, Extended Version
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We show that it is possible to locate the event horizons of a black hole (in arbitrary dimensions) as the zeros of certain Cartan invariants. This approach accounts for the recent results on the detection of stationary horizons using scalar polynomial curvature invariants, and improves upon them since the proposed method is computationally less expensive. As an application, we produce Cartan invariants that locate the event horizons for various exact four-dimensional and five-dimensional stationary, asymptotically flat (or (anti) de Sitter) black hole solutions and compare the Cartan invariants with the corresponding scalar curvature invariants that detect the event horizon. In particular, for each of the four-dimensional examples we express the scalar polynomial curvature invariants introduced by Abdelqader and Lake in terms of the Cartan invariants and show a direct relationship between the scalar polynomial curvature invariants and the Cartan invariants that detect the horizon.
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Event horizons are the defining physical features of black hole spacetimes, and are of considerable interest in studying black hole dynamics. Here, we reconsider three techniques to localise event horizons in numerical spacetimes: integrating geodesics, integrating a surface, and integrating a level-set of surfaces over a volume. We implement the first two techniques and find that straightforward integration of geodesics backward in time to be most robust. We find that the exponential rate of approach of a null surface towards the event horizon of a spinning black hole equals the surface gravity of the black hole. In head-on mergers we are able to track quasi-normal ringing of the merged black hole through seven oscillations, covering a dynamic range of about 10^5. Both at late times (when the final black hole has settled down) and at early times (before the merger), the apparent horizon is found to be an excellent approximation of the event horizon. In the head-on binary black hole merger, only {em some} of the future null generators of the horizon are found to start from past null infinity; the others approach the event horizons of the individual black holes at times far before merger.
In this paper we study the stationary horizons of the rotating black ring and the supersymmetric black ring spacetimes in five dimensions. In the case of the rotating black ring we use Weyl aligned null directions to algebraically classify the Weyl tensor, and utilize an adapted Cartan algorithm in order to produce Cartan invariants. For the supersymmetric black ring we employ the discriminant approach and repeat the adapted Cartan algorithm. For both of these metrics we are able to construct Cartan invariants that detect the horizon alone, and which are easier to compute and analyse that scalar polynomial curvature invariants.
In this article, we study Beltrami equilibria for plasmas in near the horizon of a spinning black hole, and develop a framework for constructing the magnetic field profile in the near horizon limit for Clebsch flows in the single-fluid approximation. We find that the horizon profile for the magnetic field is shown to satisfy a system of first-order coupled ODEs dependent on a boundary condition for the magnetic field. For states in which the generalized vorticity vanishes (the generalized `superconducting plasma state), the horizon profile becomes independent of the boundary condition, and depend only on the thermal properties of the plasma. Our analysis makes use of the full form for the time-independent Amperes law in the 3+1 formalism, generalizing earlier conclusions for the case of vanishing vorticity, namely the complete magnetic field expulsion near the equator of an axisymmetric black horizon assuming that the thermal properties of the plasma are symmetric about the equatorial plane. For the general case, we find and discuss additional conditions required for the expulsion of magnetic fields at given points on the black hole horizon. We perform a length scale analysis which indicates the emergence of two distinct length scales characterizing the magnetic field variation and strength of the Beltrami term, respectively.
We study curvature invariants in a binary black hole merger. It has been conjectured that one could define a quasi-local and foliation independent black hole horizon by finding the level--$0$ set of a suitable curvature invariant of the Riemann tensor. The conjecture is the geometric horizon conjecture and the associated horizon is the geometric horizon. We study this conjecture by tracing the level--$0$ set of the complex scalar polynomial invariant, $mathcal{D}$, through a quasi-circular binary black hole merger. We approximate these level--$0$ sets of $mathcal{D}$ with level--$varepsilon$ sets of $|mathcal{D}|$ for small $varepsilon$. We locate the local minima of $|mathcal{D}|$ and find that the positions of these local minima correspond closely to the level--$varepsilon$ sets of $|mathcal{D}|$ and we also compare with the level--$0$ sets of $text{Re}(mathcal{D})$. The analysis provides evidence that the level--$varepsilon$ sets track a unique geometric horizon. By studying the behaviour of the zero sets of $text{Re}(mathcal{D})$ and $text{Im}(mathcal{D})$ and also by studying the MOTSs and apparent horizons of the initial black holes, we observe that the level--$varepsilon$ set that best approximates the geometric horizon is given by $varepsilon = 10^{-3}$.
We prove the existence of general relativistic perfect fluid black hole solutions, and demonstrate the phenomenon for the $P=wrho$ class of equations of state. While admitting a local time-like Killing vector on the event horizon itself, the various black hole configurations are necessarily time dependent (thereby avoiding a well known no-go theorem) away from the horizon. Consistently, Hawkings imaginary time periodicity is globally manifest on the entire spacetime manifold.
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