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	boundary_correction_base = BoundaryCorrections::BoundaryCorrection
using	gh_system = gh::System< 3_st >
using	scalar_system = CurvedScalarWave::System< 3_st >
using	variables_tag = ::Tags::Variables< tmpl::append< typename gh_system::variables_tag::tags_list, typename scalar_system::variables_tag::tags_list > >
using	flux_variables = implementation defined
using	gradient_variables = implementation defined
using	gradients_tags = gradient_variables
using	compute_largest_characteristic_speed = Tags::ComputeLargestCharacteristicSpeed<>
using	compute_volume_time_derivative_terms = ScalarTensor::TimeDerivative
using	inverse_spatial_metric_tag = typename gh_system::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

$$\begin{array}{rl}{R}_{ab}& =8\pi T_{ab}^{(\mathrm{\Psi },\text{TR})},\\ ◻\mathrm{\Psi }& =0,\end{array}$$

where $\mathrm{\Psi }$ is the scalar field and the trace-reversed stress-energy tensor of the scalar field is given by

$$\begin{array}{rl}{T}_{ab}^{(\mathrm{\Psi },\text{TR})}& \equiv T_{ab}^{(\mathrm{\Psi })}-\frac{1}{2}{g}_{ab}{g}^{cd}{T}_{cd}^{(\mathrm{\Psi })}\\ & ={\partial }_a\mathrm{\Psi }{\partial }_b\mathrm{\Psi }.\end{array}$$

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

$$\begin{array}{rl}& g_{ab},\\ & {\mathrm{\Pi }}_{ab}=-{\displaystyle \frac{1}{\alpha }}({\partial }_t{g}_{ab}-{\beta }^k{\partial }_k{g}_{ab}),\\ & {\mathrm{\Phi }}_{iab}={\partial }_i{g}_{ab},\\ & \mathrm{\Psi },\\ & \mathrm{\Pi }=-{\displaystyle \frac{1}{\alpha }}({\partial }_t\mathrm{\Psi }-{\beta }^k{\partial }_k\mathrm{\Psi }),\\ & {\mathrm{\Phi }}_i={\partial }_i\mathrm{\Psi },\end{array}$$

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.

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

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