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 {}