Line data    Source code

       1            1 : // Distributed under the MIT License.
       2              : // See LICENSE.txt for details.
       3              : 
       4              : /// \file
       5              : /// Documents the `Xcts` namespace
       6              : 
       7              : #pragma once
       8              : 
       9              : /*!
      10              :  * \ingroup EllipticSystemsGroup
      11              :  * \brief Items related to solving the Extended Conformal Thin Sandwich (XCTS)
      12              :  * decomposition of the Einstein constraint equations
      13              :  *
      14              :  * The XCTS equations
      15              :  *
      16              :  * \f{align}
      17              :  * \bar{D}^2 \psi - \frac{1}{8}\psi\bar{R} - \frac{1}{12}\psi^5 K^2 +
      18              :  * \frac{1}{8}\psi^{-7}\bar{A}_{ij}\bar{A}^{ij} &= -2\pi\psi^5\rho
      19              :  * \\
      20              :  * \bar{D}_i(\bar{L}\beta)^{ij} - (\bar{L}\beta)^{ij}\bar{D}_i
      21              :  * \ln(\bar{\alpha}) &= \bar{\alpha}\bar{D}_i\left(\bar{\alpha}^{-1}\bar{u}^{ij}
      22              :  * \right) + \frac{4}{3}\bar{\alpha}\psi^6\bar{D}^j K + 16\pi\bar{\alpha}
      23              :  * \psi^{10}S^j
      24              :  * \\
      25              :  * \bar{D}^2\left(\alpha\psi\right) &=
      26              :  * \alpha\psi\left(\frac{7}{8}\psi^{-8}\bar{A}_{ij}\bar{A}^{ij}
      27              :  * + \frac{5}{12}\psi^4 K^2 + \frac{1}{8}\bar{R}
      28              :  * + 2\pi\psi^4\left(\rho + 2S\right)\right)
      29              :  * - \psi^5\partial_t K + \psi^5\beta^i\bar{D}_i K
      30              :  * \\
      31              :  * \text{with} \quad \bar{A} &= \frac{1}{2\bar{\alpha}}
      32              :  * \left((\bar{L}\beta)^{ij} - \bar{u}^{ij}\right) \\
      33              :  * \quad \text{and} \quad \bar{\alpha} &= \alpha \psi^{-6}
      34              :  * \f}
      35              :  *
      36              :  * are a set of nonlinear elliptic equations that the spacetime metric in
      37              :  * general relativity must satisfy at all times. For an introduction see e.g.
      38              :  * \cite BaumgarteShapiro, in particular Box 3.3 which is largely mirrored here.
      39              :  * We solve the XCTS equations for the conformal factor \f$\psi\f$, the product
      40              :  * of lapse times conformal factor \f$\alpha\psi\f$ and the shift vector
      41              :  * \f$\beta^j\f$. The remaining quantities in the equations, i.e. the conformal
      42              :  * metric \f$\bar{\gamma}_{ij}\f$, the trace of the extrinsic curvature \f$K\f$,
      43              :  * their respective time derivatives \f$\bar{u}_{ij}\f$ and \f$\partial_t K\f$,
      44              :  * the energy density \f$\rho\f$, the stress-energy trace \f$S\f$ and the
      45              :  * momentum density \f$S^i\f$, are freely specifyable fields that define the
      46              :  * physical scenario at hand. Of particular importance is the conformal metric,
      47              :  * which defines the background geometry, the covariant derivative
      48              :  * \f$\bar{D}\f$, the Ricci scalar \f$\bar{R}\f$ and the longitudinal operator
      49              :  *
      50              :  * \f{equation}
      51              :  * \left(\bar{L}\beta\right)^{ij} = \bar{D}^i\beta^j + \bar{D}^j\beta^i
      52              :  * - \frac{2}{3}\bar{\gamma}^{ij}\bar{D}_k\beta^k
      53              :  * \text{.}
      54              :  * \f}
      55              :  *
      56              :  * Note that the XCTS equations are essentially two Poisson equations and one
      57              :  * Elasticity equation with nonlinear sources on a curved geometry. In this
      58              :  * analogy, the longitudinal operator plays the role of the elastic constitutive
      59              :  * relation that connects the symmetric "shift strain"
      60              :  * \f$\bar{D}_{(i}\beta_{j)}\f$ with the "stress" \f$(\bar{L}\beta)^{ij}\f$ of
      61              :  * which we take the divergence in the momentum constraint. This particular
      62              :  * constitutive relation is equivalent to an isotropic and homogeneous material
      63              :  * with bulk modulus \f$K=0\f$ (not to be confused with the extrinsic curvature
      64              :  * trace \f$K\f$ in this context) and shear modulus \f$\mu=1\f$ (see
      65              :  * `Elasticity::ConstitutiveRelations::IsotropicHomogeneous`).
      66              :  *
      67              :  * Once the XCTS equations are solved we can construct the spatial metric and
      68              :  * extrinsic curvature as
      69              :  *
      70              :  * \f{align}
      71              :  * \gamma_{ij} &= \psi^4\bar{\gamma}_{ij} \\
      72              :  * K_{ij} &= \psi^{-2}\bar{A}_{ij} + \frac{1}{3}\gamma_{ij} K
      73              :  * \f}
      74              :  *
      75              :  * from which we can compose the full spacetime metric.
      76              :  */
      77              : namespace Xcts {}