ScalarTensor::System Struct Reference

Scalar Tensor system obtained from combining the CurvedScalarWave and gh systems. More...

#include <System.hpp>

Public Types
using	boundary_conditions_base = BoundaryConditions::BoundaryCondition
using	gh_system = gh::System<3_st>
using	scalar_system = CurvedScalarWave::System<3_st>
using	variables_tag
using	flux_variables
using	gradient_variables
using	gradients_tags = gradient_variables
using	compute_largest_characteristic_speed
using	compute_volume_time_derivative_terms = ScalarTensor::TimeDerivative
using	inverse_spatial_metric_tag

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

Detailed Description

Scalar Tensor system obtained from combining the CurvedScalarWave and gh systems.

Details

The evolution equations follow from

\begin{align*} R_{ab} &= 8 \pi \, T^{(\Psi, \text{TR})}_{ab} ~, \\ \Box \Psi &= 0~, \end{align*}

where \(\Psi\) is the scalar field and the trace-reversed stress-energy tensor of the scalar field is given by

\begin{align*} T^{(\Psi, \text{TR})}_{ab} &\equiv T^{(\Psi)}_{ab} - \frac{1}{2} g_{ab} g^{cd} T^{(\Psi)}_{cd} \\ &= \partial_a \Psi \partial_b \Psi ~. \end{align*}

Both systems are recast as first-order systems in terms of the variables

\begin{align*}& g_{ab}~, \\ & \Pi_{ab} = - \dfrac{1}{\alpha} \left( \partial_t g_{ab} - \beta^k \partial_k g_{ab} \right)~, \\ & \Phi_{iab} = \partial_i g_{ab}~, \\ & \Psi~, \\ & \Pi = - \dfrac{1}{\alpha} \left(\partial_t \Psi - \beta^k \partial_k \Psi \right)~, \\ & \Phi_i = \partial_i \Psi~, \end{align*}

where \( \alpha \) and \( \beta^k \) are the lapse and shift.

The computation of the evolution equations is implemented in each system in gh::TimeDerivative and CurvedScalarWave::TimeDerivative, respectively. We take the additional step of adding the contribution of the trace-reversed stress-energy tensor to the evolution equations of the metric.

Note

Although both systems are templated in the spatial dimension, we only implement this system in three spatial dimensions.

Member Typedef Documentation

compute_largest_characteristic_speed

using ScalarTensor::System::compute_largest_characteristic_speed

Initial value:

 
      Tags::ComputeLargestCharacteristicSpeed<>

flux_variables

using ScalarTensor::System::flux_variables

Initial value:

 tmpl::append<typename gh_system::flux_variables,
                                      typename scalar_system::flux_variables>

gradient_variables

using ScalarTensor::System::gradient_variables

Initial value:

 
      tmpl::append<typename gh_system::gradient_variables,
                   typename scalar_system::gradient_variables>

inverse_spatial_metric_tag

using ScalarTensor::System::inverse_spatial_metric_tag

Initial value:

 
      typename gh_system::inverse_spatial_metric_tag

variables_tag

using ScalarTensor::System::variables_tag

Initial value:

 ::Tags::Variables<
      tmpl::append<typename gh_system::variables_tag::tags_list,
                   typename scalar_system::variables_tag::tags_list>>

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

• src/Evolution/Systems/ScalarTensor/System.hpp