SpECTRE
v2024.05.11
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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 = tmpl::append< typename gh_system::flux_variables, typename scalar_system::flux_variables > |
using | gradient_variables = tmpl::append< typename gh_system::gradient_variables, typename scalar_system::gradient_variables > |
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 |
Scalar Tensor system obtained from combining the CurvedScalarWave and gh systems.
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.