Cce::KleinGordonSystem< EvolveCcm > Struct Template Reference

Performing Cauchy characteristic evolution and Cauchy characteristic matching for Einstein-Klein-Gordon system. More...

#include <KleinGordonSystem.hpp>

Public Types
using	variables_tag = implementation defined

Static Public Attributes
static constexpr size_t	volume_dim = 3
static constexpr bool	has_primitive_and_conservative_vars = false

Detailed Description

template<bool EvolveCcm>
struct Cce::KleinGordonSystem< EvolveCcm >

Performing Cauchy characteristic evolution and Cauchy characteristic matching for Einstein-Klein-Gordon system.

Details

The code adopts the characteristic formulation to solve the field equations for scalar-tensor theory as considered in [136]. 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:

$$S=\int d^{4}x\sqrt{-g}(\frac{R}{16\pi }-\frac{1}{2}{\mathrm{\nabla }}_{\mu }\psi {\mathrm{\nabla }}^{\mu }\psi ).$$

The system consists of two sectors: scalar and tensor (metric). The scalar field follows the Klein-Gordon (KG) equation

$$◻\psi =0.$$

Its characteristic expression is given in [12], yielding the hypersurface equation for ${\partial }_u\psi =\mathrm{\Pi }$, 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 $\mathrm{\Pi }$. 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 $(\mathrm{\Pi })$. 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:

• Integrate the metric hypersurface equations with the existing infrastructure
• Integrate the KG equation for $\mathrm{\Pi }$
• Evolve two variables, $\psi$ (scalar) and $J$ (tensor) to the next time step

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The documentation for this struct was generated from the following file:

• src/Evolution/Systems/Cce/KleinGordonSystem.hpp