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Finding Black Holes 1

时间:2019-09-17 17:38:58      阅读:70      评论:0      收藏:0      [点我收藏+]

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  • Finding Black Holes 1

    The apparent horizon (i.e., the marginally trapped outer surface) is an invaluable tool for finding black holes in
    numerical relativity: In numerical relativity, the existence of a black hole is usually confirmed by finding the presence of an apparent horizon.
    By contrast to the event horizon that is related to a global structure of spacetime, the apparent horizon can be defined on each spatial hypersurface \(\Sigma_t\).
    We denote an future-directed outgoing null vector field as \(k^a\) and suppose that it is tangent of null geodesics. Then, we have the relations
    \[ k^ak_a =0,\text{and } k^a\nabla_b k^a=0 \]
    Defining another null vector field, \(\ell^a\), such that \(k^a\ell_a=-1\), the spacetime metric is written as
    \[ g_{ab}=-k_a\ell_b-\ell_ak_b+H_{ab} \]
    where \(H_{ab}=\gamma_{ab}-s_as_b\) is a two-dimensional metric that satisfies \(H_{ab}k^a = H_{ab}\ell^a = 0.\)
    the expansion \[\Theta=H^{ab}\nabla_ak_b\]
    \[ \begin{align} \Theta &=H^{ab}\nabla_ak_b=0\ &=(\gamma^{ab}-s^as^b) \nabla_ak_b=0\ &=D_as^a+K_{ab}s^as^b-K=0 \end{align} \]
    The next task is to rewrite equation \(D_as^a+K_{ab}s^as^b-K=0\) to a form by which the surface of an apparent horizon can be located.
    For this purpose, we denote the surface of the apparent horizon by
    \[ r = f(\theta_k)\]
    where \(f\) is a function to be determined and \(\theta_k (k = 1,2,..N-1)\) denotes a set of angular coordinates of the apparent horizon
    (remember we assume that the apparent horizon has a spherical topology).
    \[ \begin{align} s_i &=C\nabla_i(r-f(\theta_k))=C(1,\partial_i f), i\neq r\ C &=(\gamma^{rr}-2\gamma^{rj}\partial_j f+\gamma^{jk}\partial_j f\partial_k f)^{-1/2} \end{align} \]
    We will assume that spherical polar coordinates \((r,\theta,\phi)\) are used in the following. \((N=3)\)

Finding Black Holes 1

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原文地址:https://www.cnblogs.com/yuewen-chen/p/11535396.html

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