Line data    Source code

       1           1 : // Distributed under the MIT License.
       2             : // See LICENSE.txt for details.
       3             : 
       4             : /// \file
       5             : /// Documents the `Punctures` namespace
       6             : 
       7             : #pragma once
       8             : 
       9             : /*!
      10             :  * \ingroup EllipticSystemsGroup
      11             :  * \brief Items related to solving the puncture equation
      12             :  *
      13             :  * The puncture equation
      14             :  *
      15             :  * \begin{equation}\label{eq:puncture_eqn}
      16             :  * -\nabla^2 u = \beta \left(\alpha \left(1 + u\right) + 1\right)^{-7}
      17             :  * \end{equation}
      18             :  *
      19             :  * is a nonlinear Poisson-type elliptic PDE for the "puncture field" $u$. See
      20             :  * Eq. (12.52) and surrounding discussion in \cite BaumgarteShapiro, or
      21             :  * \cite BrandtBruegmann1997 for an introduction. To arrive at the puncture
      22             :  * equation we assume conformal flatness and maximal slicing in vacuum so the
      23             :  * Einstein momentum constraint becomes homogeneous:
      24             :  *
      25             :  * \begin{equation}\label{eq:mom_constraint}
      26             :  * \nabla_j \bar{A}^{ij} = 0
      27             :  * \end{equation}
      28             :  *
      29             :  * Here, $\nabla$ is the flat-space covariant derivate. $\bar{A}^{ij}$ is the
      30             :  * conformal traceless extrinsic curvature that composes the extrinsic curvature
      31             :  * as
      32             :  *
      33             :  * \begin{equation}
      34             :  * K_{ij} = \psi^{-2} \bar{A}_{ij} + \frac{1}{3} \gamma_{ij} K
      35             :  * \end{equation}
      36             :  *
      37             :  * (where $K=0$ under maximal slicing). $\psi$ is the conformal factor that
      38             :  * composes the spatial metric as
      39             :  *
      40             :  * \begin{equation}
      41             :  * \gamma_{ij} = \psi^4 \bar{\gamma}_{ij}
      42             :  * \end{equation}
      43             :  *
      44             :  * (where $\bar{\gamma}_{ij} = \delta{ij}$ under conformal flatness and in
      45             :  * Cartesian coordinates).
      46             :  *
      47             :  * The momentum constraint ($\ref{eq:mom_constraint}$) is solved analytically by
      48             :  * the Bowen-York extrinsic curvature
      49             :  *
      50             :  * \begin{equation}
      51             :  * \bar{A}^{ij} = \frac{3}{2} \frac{1}{r_C^2} \left(
      52             :  * 2 P^{(i} n^{j)} - (\delta^{ij} - n^i n^j) P^k n^k
      53             :  * + \frac{4}{r_C} n^{(i} \epsilon^{j)kl} S^k n^l\right)
      54             :  * \end{equation}
      55             :  *
      56             :  * representing a black hole with linear momentum $\mathbf{P}$ and angular
      57             :  * momentum $\mathbf{S}$ at position $\mathbf{C}$. The quantity
      58             :  * $r_C=||\mathbf{x}-\mathbf{C}||$ is the Euclidean coordinate distance to the
      59             :  * black hole, and $\mathbf{n}=(\mathbf{x}-\mathbf{C})/r_C$ is the radial unit
      60             :  * normal to the black hole. Since the momentum constraint is linear, any
      61             :  * superposition of $\bar{A}^{ij}$ is also a solution to the momentum
      62             :  * constraint, allowing to represent multiple black holes.
      63             :  *
      64             :  * Only the Einstein Hamiltonian constraint remains to be solved numerically for
      65             :  * the conformal factor:
      66             :  *
      67             :  * \begin{equation}
      68             :  * \nabla^2 \psi = \frac{1}{8} \psi^{-7} \bar{A}_{ij} \bar{A}^{ij}
      69             :  * \end{equation}
      70             :  *
      71             :  * It reduces to the puncture equation ($\ref{eq:puncture_eqn}$) when we
      72             :  * decompose the conformal factor as:
      73             :  *
      74             :  * \begin{equation}
      75             :  * \psi = 1 + \frac{1}{\alpha} + u
      76             :  * \end{equation}
      77             :  *
      78             :  * where we define
      79             :  *
      80             :  * \begin{equation}
      81             :  * \frac{1}{\alpha} = \sum_I \frac{M_I}{2 r_I}
      82             :  * \end{equation}
      83             :  *
      84             :  * and
      85             :  *
      86             :  * \begin{equation}
      87             :  * \beta = \frac{1}{8} \alpha^7 \bar{A}_{ij} \bar{A}^{ij}.
      88             :  * \end{equation}
      89             :  *
      90             :  * Here, $M_I$ is the "puncture mass" (or "bare mass") parameter for the $I$th
      91             :  * black hole at position $\mathbf{C}_I$, and $\bar{A}_{ij}$ is the
      92             :  * superposition of the Bowen-York extrinsic curvature of the black holes with
      93             :  * the parameters defined above. Note that the definition of $\frac{1}{\alpha}$
      94             :  * in Eq. (12.51) in \cite BaumgarteShapiro is missing factors of $\frac{1}{2}$,
      95             :  * but their Eq. (3.23) includes them, as does Eq. (8) in
      96             :  * \cite BrandtBruegmann1997 (though the latter includes the unit offset in
      97             :  * the definition of $u$).
      98             :  */
      99             : namespace Punctures {}