SpECTRE
v2024.02.05

Performing Cauchy characteristic evolution and Cauchy characteristic matching for EinsteinKleinGordon 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 EinsteinKleinGordon system.
The code adopts the characteristic formulation to solve the field equations for scalartensor theory as considered in [123]. Working in the Einstein frame, a realvalued 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 KleinGordon (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 EinsteinKleinGordon system can be divided into three major sequential steps: