Xcts.hpp
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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 {}
Xcts
Items related to solving the Extended Conformal Thin Sandwich (XCTS) decomposition of the Einstein co...
Definition: Flatness.hpp:22