Line data    Source code

       1            1 : // Distributed under the MIT License.
       2              : // See LICENSE.txt for details.
       3              : 
       4              : /// \file
       5              : /// Documents the `sgb` namespace
       6              : 
       7              : #pragma once
       8              : 
       9              : /*!
      10              :  * \ingroup EllipticSystemsGroup
      11              :  * \brief Items related to solving the sGB scalar equation
      12              :  *
      13              :  * The quasi-stationary scalar equation in sGB gravity
      14              :  * \begin{equation}\label{eq:sGB}
      15              :  * -\partial_i \left[ \left( \gamma^{ij} - \alpha^{-2} \beta^i \beta^j \right)
      16              :  * \partial_j \Psi \right] + \partial_j \Psi \left( \gamma^{ij} - \alpha^{-2}
      17              :  * \beta^i \beta^j \right) \left( \Gamma_i + \partial_i \ln \alpha \right) =
      18              :  * \ell^2 f' \left( \Psi \right) \mathcal{G}
      19              :  * \end{equation}
      20              :  *
      21              :  * is a nonlinear Poisson-type elliptic PDE for the scalar field $\Psi$. To
      22              :  * obtain this equation, one begins by considering the action:
      23              :  *
      24              :  * \begin{equation}
      25              :  * S\left[g_{ab}, \Psi \right] \equiv \int \, d^4 x \sqrt{-g}
      26              :  * \Big[ \dfrac{R}{2 \kappa } - \dfrac{1}{2}  \nabla_{a} \Psi \nabla^{a} \Psi +
      27              :  * \ell^2 f(\Psi) \, \mathcal{G} \Big],
      28              :  * \end{equation}
      29              :  *
      30              :  * where $\mathcal{G} \equiv R_{abcd}R^{abcd} - 4 R_{ab}R^{ab} + R^2$. Varying
      31              :  * the action with respect to $\Psi$, one obtains the wave-like equation
      32              :  *
      33              :  * \begin{equation}
      34              :  * \Box \Psi = - \ell^2 f'(\Psi) \mathcal{G}
      35              :  * \end{equation}
      36              :  *
      37              :  * In the spirit of quasi-stationarity, we set $\partial_t \Psi = \partial_t^2
      38              :  * \Psi = \partial_t \alpha = \partial_t \beta^{i} = 0$, where $\alpha$ and
      39              :  * $\beta^i$ are the lapse and shift respectively. This yields ($\ref{eq:sGB}$),
      40              :  * with $\gamma^{ij}$ being the spatial metric, and $\Gamma_i$ its associated
      41              :  * contracted christoffel symbol of the second kind.
      42              :  *
      43              :  * Currently, we have implemented the coupling function
      44              :  *
      45              :  * \begin{equation}
      46              :  * \ell^2 f(\Psi) = \epsilon_2 \frac{\Psi^2}{8} + \epsilon_4 \frac{\Psi^4}{16}
      47              :  * \end{equation}
      48              :  *
      49              :  * Note that the principal part of the master equation will generically turn
      50              :  * singular at black hole horizon's for stationary initial data. Typically, this
      51              :  * requires a boundary condition ensuring regularity of the solution. However,
      52              :  * as detailed in \cite Nee2024bur, this is already enforced by the chosen
      53              :  * spectral decomposition, and so instead one should impose the DoNothing
      54              :  * boundary condition for excision surfaces within black hole apparent horizons.
      55              :  *
      56              :  * As is currently implemented, one must provide numeric data corresponding to
      57              :  * the full metric $g_{ab}$. All of this can be generated using SolveXcts,
      58              :  * with a glob for the volume files being specified in the
      59              :  * SolveScalarGaussBonnet input file.
      60              :  *
      61              :  * Specifically, one must ensure the following is in the provided volume file:
      62              :  *
      63              :  * - the conformal factor $\psi$,
      64              :  * - the spatial metric $\gamma_{ij}$,
      65              :  * - the lapse $\alpha$,
      66              :  * - the shift $\beta^i$,
      67              :  * - the shift excess $\beta_{exc}^i$,
      68              :  * - the extrinsic curvature $K^{ij}$, and
      69              :  * - the inverse conformal metric $\bar{\gamma}^{ij}$.
      70              :  *
      71              :  * One must also ensure that the Background specified in the input file is the
      72              :  * same one that was used in SolveXcts. This is checked using
      73              :  * $\bar{\gamma}^{ij}$.
      74              :  */
      75              : namespace sgb {}