tensorium_RG::BSSNAtildeTensor< K > Class Template Reference

Computes the trace-free conformal extrinsic curvature tensor \( \tilde{A}_{ij} \) in the BSSN formalism. More...

#include <BSSNAtildeTensor.hpp>

Collaboration diagram for tensorium_RG::BSSNAtildeTensor< K >:

Public Types
using	Vec = tensorium::Vector<K>
using	Mat = tensorium::Tensor<K, 2>

Public Member Functions
tensorium::Tensor< K, 2 >	compute_Atilde_tensor (const tensorium::Tensor< K, 2 > &Kij, const tensorium::Tensor< K, 2 > &gamma_inv, const tensorium::Tensor< K, 2 > &gamma, K chi)
	Computes \( \tilde{A}_{ij} \) from \( K_{ij} \), \( \gamma_{ij} \), and \( \chi \).

Detailed Description

template<typename K>
class tensorium_RG::BSSNAtildeTensor< K >

Computes the trace-free conformal extrinsic curvature tensor \( \tilde{A}_{ij} \) in the BSSN formalism.

This tensor is defined as:

\[ \tilde{A}_{ij} = \chi \left( K_{ij} - \frac{1}{3} \gamma_{ij} K \right) \]

where:

• \( K_{ij} \) is the extrinsic curvature tensor
• \( K = \gamma^{ij} K_{ij} \) is its trace
• \( \chi \) is the conformal factor
• \( \gamma_{ij} \) is the spatial 3-metric

Member Typedef Documentation

Mat

template<typename K >

using tensorium_RG::BSSNAtildeTensor< K >::Mat = tensorium::Tensor<K, 2>

Vec

template<typename K >

using tensorium_RG::BSSNAtildeTensor< K >::Vec = tensorium::Vector<K>

Member Function Documentation

compute_Atilde_tensor()

template<typename K >

tensorium::Tensor< K, 2 > tensorium_RG::BSSNAtildeTensor< K >::compute_Atilde_tensor ( const tensorium::Tensor< K, 2 > & Kij, const tensorium::Tensor< K, 2 > & gamma_inv, const tensorium::Tensor< K, 2 > & gamma, K chi )

inline

Computes \( \tilde{A}_{ij} \) from \( K_{ij} \), \( \gamma_{ij} \), and \( \chi \).

\[ \tilde{A}_{ij} = \chi \left( K_{ij} - \frac{1}{3} \gamma_{ij} K \right), \quad K = \gamma^{ij} K_{ij} \]

Parameters

Kij	Extrinsic curvature tensor \( K_{ij} \)
gamma_inv	Inverse spatial metric \( \gamma^{ij} \)
gamma	Spatial metric \( \gamma_{ij} \)
chi	Conformal factor \( \chi \)

Returns

Trace-free conformal extrinsic curvature tensor \( \tilde{A}_{ij} \)

References chi, and gamma.

Referenced by tensorium_RG::BSSN< T >::init_BSSN().

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

• includes/Tensorium/DiffGeometry/BSSN/BSSNAtildeTensor.hpp