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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 {}