Finding Black Holes 1

• 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).
$$
\begin{align}
s_i &=C\nabla_i(r-f(\theta_k))=C(1,\partial_i f), i\neq r\
C &=(\gamma^{rr}-2\gamma\partial_j f+\gamma^{jk}\partial_j f\partial_k f)^{-1/2}
\end{align}
$$
We will assume that spherical polar coordinates \((r,\theta,\phi)\) are used in the following. \((N=3)\)