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src
Elliptic
Systems
Xcts
Xcts.hpp
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// Distributed under the MIT License.
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// See LICENSE.txt for details.
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/// \file
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/// Documents the `Xcts` namespace
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#pragma once
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/*!
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* \ingroup EllipticSystemsGroup
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* \brief Items related to solving the Extended Conformal Thin Sandwich (XCTS)
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* decomposition of the Einstein constraint equations
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*
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* The XCTS equations
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*
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* \f{align}
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* \bar{D}^2 \psi - \frac{1}{8}\psi\bar{R} - \frac{1}{12}\psi^5 K^2 +
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* \frac{1}{8}\psi^{-7}\bar{A}_{ij}\bar{A}^{ij} &= -2\pi\psi^5\rho
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* \\
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* \bar{D}_i(\bar{L}\beta)^{ij} - (\bar{L}\beta)^{ij}\bar{D}_i
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* \ln(\bar{\alpha}) &= \bar{\alpha}\bar{D}_i\left(\bar{\alpha}^{-1}\bar{u}^{ij}
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* \right) + \frac{4}{3}\bar{\alpha}\psi^6\bar{D}^j K + 16\pi\bar{\alpha}
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* \psi^{10}S^j
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* \\
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* \bar{D}^2\left(\alpha\psi\right) &=
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* \alpha\psi\left(\frac{7}{8}\psi^{-8}\bar{A}_{ij}\bar{A}^{ij}
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* + \frac{5}{12}\psi^4 K^2 + \frac{1}{8}\bar{R}
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* + 2\pi\psi^4\left(\rho + 2S\right)\right)
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* - \psi^5\partial_t K + \psi^5\beta^i\bar{D}_i K
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* \\
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* \text{with} \quad \bar{A} &= \frac{1}{2\bar{\alpha}}
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* \left((\bar{L}\beta)^{ij} - \bar{u}^{ij}\right) \\
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* \quad \text{and} \quad \bar{\alpha} &= \alpha \psi^{-6}
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* \f}
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*
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* are a set of nonlinear elliptic equations that the spacetime metric in
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* general relativity must satisfy at all times. For an introduction see e.g.
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* \cite BaumgarteShapiro, in particular Box 3.3 which is largely mirrored here.
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* We solve the XCTS equations for the conformal factor \f$\psi\f$, the product
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* of lapse times conformal factor \f$\alpha\psi\f$ and the shift vector
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* \f$\beta^j\f$. The remaining quantities in the equations, i.e. the conformal
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* metric \f$\bar{\gamma}_{ij}\f$, the trace of the extrinsic curvature \f$K\f$,
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* their respective time derivatives \f$\bar{u}_{ij}\f$ and \f$\partial_t K\f$,
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* the energy density \f$\rho\f$, the stress-energy trace \f$S\f$ and the
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* momentum density \f$S^i\f$, are freely specifyable fields that define the
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* physical scenario at hand. Of particular importance is the conformal metric,
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* which defines the background geometry, the covariant derivative
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* \f$\bar{D}\f$, the Ricci scalar \f$\bar{R}\f$ and the longitudinal operator
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*
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* \f{equation}
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* \left(\bar{L}\beta\right)^{ij} = \bar{D}^i\beta^j + \bar{D}^j\beta^i
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* - \frac{2}{3}\bar{\gamma}^{ij}\bar{D}_k\beta^k
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* \text{.}
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* \f}
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*
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* Note that the XCTS equations are essentially two Poisson equations and one
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* Elasticity equation with nonlinear sources on a curved geometry. In this
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* analogy, the longitudinal operator plays the role of the elastic constitutive
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* relation that connects the symmetric "shift strain"
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* \f$\bar{D}_{(i}\beta_{j)}\f$ with the "stress" \f$(\bar{L}\beta)^{ij}\f$ of
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* which we take the divergence in the momentum constraint. This particular
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* constitutive relation is equivalent to an isotropic and homogeneous material
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* with bulk modulus \f$K=0\f$ (not to be confused with the extrinsic curvature
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* trace \f$K\f$ in this context) and shear modulus \f$\mu=1\f$ (see
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* `Elasticity::ConstitutiveRelations::IsotropicHomogeneous`).
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*
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* Once the XCTS equations are solved we can construct the spatial metric and
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* extrinsic curvature as
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*
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* \f{align}
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* \gamma_{ij} &= \psi^4\bar{\gamma}_{ij} \\
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* K_{ij} &= \psi^{-2}\bar{A}_{ij} + \frac{1}{3}\gamma_{ij} K
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* \f}
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*
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* from which we can compose the full spacetime metric.
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*/
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namespace
Xcts
{}
Xcts
Items related to solving the Extended Conformal Thin Sandwich (XCTS) decomposition of the Einstein co...
Definition:
Flatness.hpp:22
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