Brill-Lindquist

The Brill-Lindquist initial data is an alternate form of multi-throat data which differs from the Misner data mainly in its choice of spacetime topology. Whereas the Misner data presumes that the throats connect a pair of asymptotically flat spacetimes which are identical to each other, the Brill-Lindquist data connects each throat to a separate asymptotically flat region [36].

Figure D12.3: Two Brill-Lindquist throats connecting separate asymptotically flat regions.

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The form of the conformal factor is:

$$\psi = 1 + \sum_{i=1}^N \frac{m_i}{2r_i},$$ (part3220)

where the $m_i$ and $r_i$ are the masses and positions of the $i$ particles.

The parameter specifying the number of black holes is idanalyticbh::bl_nbh. A maximum of four black holes can be specified. The mass and $(x,y,z)$ position of the first black hole is given by bl_M_1, bl_x0_1, bl_y0_1, bl_z0_1, with corresponding parameters for the second to fourth black holes. Note that the default values for each of the position coordinates are $0.0$, so that only the coordinates off of the axes must be specified.

If the conformal metric is used, then derivatives of the conformal factor are calculated from the analytic derivatives of the above expression for the conformal factor.

To initialise a run with a pair of Brill-Lindquist black holes with masses $1$ and $2$ and located at $\pm 1$ on the $y$-axis, a set of parameters such as the following could be used:

  ActiveThorns = "... ADMBase StaticConformal IDAnalyticBH ..."

  admbase::metric_type = "static conformal"

  admbase::initial_data = "bl_bh"

  idanalyticbh::bl_nbh = 2

  idanalyticbh::bl_M_1 = 1.0
  idanalyticbh::bl_y0_1 = 1.0

  idanalyticbh::bl_M_2 = 2.0
  idanalyticbh::bl_y0_2 = -1.0