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 {}