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SpECTRE
v2024.09.29
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Performing Cauchy characteristic evolution and Cauchy characteristic matching for Einstein-Klein-Gordon system. More...
#include <KleinGordonSystem.hpp>
Public Types | |
| using | variables_tag = tmpl::list< ::Tags::Variables< tmpl::list< Tags::BondiJ, Tags::KleinGordonPsi > >, ::Tags::Variables< tmpl::conditional_t< EvolveCcm, tmpl::list< Cce::Tags::CauchyCartesianCoords, Cce::Tags::PartiallyFlatCartesianCoords, Cce::Tags::InertialRetardedTime >, tmpl::list< Cce::Tags::CauchyCartesianCoords, Cce::Tags::InertialRetardedTime > > > > |
Static Public Attributes | |
| static constexpr size_t | volume_dim = 3 |
| static constexpr bool | has_primitive_and_conservative_vars = false |
Performing Cauchy characteristic evolution and Cauchy characteristic matching for Einstein-Klein-Gordon system.
The code adopts the characteristic formulation to solve the field equations for scalar-tensor theory as considered in [132]. Working in the Einstein frame, a real-valued scalar field \(\psi\) is minimally coupled with the spacetime metric \(g_{\mu\nu}\). The corresponding action is expressed as follows:
\[ S = \int d^4x \sqrt{-g} \left(\frac{R}{16 \pi} - \frac{1}{2} \nabla_\mu \psi \nabla^\mu \psi\right). \]
The system consists of two sectors: scalar and tensor (metric). The scalar field follows the Klein-Gordon (KG) equation
\[ \Box \psi = 0. \]
Its characteristic expression is given in [12], yielding the hypersurface equation for \(\partial_u\psi=\Pi\), where \(\partial_u\) represents differentiation with respect to retarded time \(u\) at fixed numerical radius \(y\). The code first integrates the KG equation radially to determine \(\Pi\). Subsequently, the time integration is performed to evolve the scalar field \(\psi\) forward in time.
The tensor (metric) sector closely aligns with the current GR CCE system, incorporating additional source terms that depend only on the scalar field \(\psi\) and its spatial derivatives, rather than its time derivative \((\Pi)\). This feature preserves the hierarchical structure of the equations. As a result, the Einstein-Klein-Gordon system can be divided into three major sequential steps: