Finding Black Holes 1

Posted 2020-11-13 yuewen-chen

• 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, ext{and } k^a abla_b k^a=0 ]
  Defining another null vector field, (ell^a), such that (k^aell_a=-1), the spacetime metric is written as
  [ g_{ab}=-k_aell_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} abla_ak_b]
  [ egin{align} Theta &=H^{ab} abla_ak_b=0 &=(gamma^{ab}-s^as^b) abla_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( heta_k)]
  where (f) is a function to be determined and ( heta_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).
  [ egin{align} s_i &=C abla_i(r-f( heta_k))=C(1,partial_i f), i eq r C &=(gamma^{rr}-2gamma^{rj}partial_j f+gamma^{jk}partial_j fpartial_k f)^{-1/2} end{align} ]
  We will assume that spherical polar coordinates ((r, heta,phi)) are used in the following. ((N=3))