tensorium_RG::ExtrinsicCurvature< K > Class Template Reference

Computes the extrinsic curvature tensor \( K_{ij} \) from BSSN variables. More...

#include <BSSNextrinTensor.hpp>

Collaboration diagram for tensorium_RG::ExtrinsicCurvature< K >:

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

Public Member Functions
template<typename T >
tensorium::Tensor< T, 2 >	compute_Kij (const tensorium::Tensor< T, 2 > &dgt, const tensorium::Tensor< T, 2 > &gamma, const tensorium::Vector< T > &beta, const tensorium::Tensor< T, 2 > &partial_beta, const tensorium::Tensor< T, 3 > &christoffel, const T alpha)
	Computes the extrinsic curvature tensor \( K_{ij} \).

Detailed Description

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

Computes the extrinsic curvature tensor \( K_{ij} \) from BSSN variables.

The extrinsic curvature is computed from the Lie derivative of the metric with respect to the shift vector and the lapse function according to the BSSN equation:

\[ K_{ij} = -\frac{1}{2\alpha} \left( \partial_t \gamma_{ij} - \nabla_i \beta_j - \nabla_j \beta_i \right) \]

where:

• \( \gamma_{ij} \) is the spatial metric
• \( \alpha \) is the lapse function
• \( \beta^i \) is the shift vector
• \( \nabla_i \beta_j = \partial_i \beta_j - \Gamma^k_{ij} \beta_k \)

The implementation uses:

• dgt = \( \partial_t \gamma_{ij} \)
• partial_beta(i, j) = \( \partial_i \beta_j \)
• christoffel(i, j, k) = \( \Gamma^k_{ij} \)

Template Parameters

K	The scalar type used (e.g., double)

Member Typedef Documentation

Mat

template<typename K >

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

Vec

template<typename K >

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

Member Function Documentation

compute_Kij()

template<typename K >

template<typename T >

tensorium::Tensor< T, 2 > tensorium_RG::ExtrinsicCurvature< K >::compute_Kij ( const tensorium::Tensor< T, 2 > & dgt, const tensorium::Tensor< T, 2 > & gamma, const tensorium::Vector< T > & beta, const tensorium::Tensor< T, 2 > & partial_beta, const tensorium::Tensor< T, 3 > & christoffel, const T alpha )

inline

Computes the extrinsic curvature tensor \( K_{ij} \).

\[ K_{ij} = -\frac{1}{2\alpha} \left( \partial_t \gamma_{ij} - (\partial_i \beta_j + \partial_j \beta_i - 2 \Gamma^k_{ij} \beta_k) \right) \]

The result is symmetrized to ensure \( K_{ij} = K_{ji} \).

Parameters

dgt	Time derivative \( \partial_t \gamma_{ij} \)
gamma	The 3×3 spatial metric \( \gamma_{ij} \)
beta	The shift vector \( \beta^i \)
partial_beta	The partial derivatives \( \partial_i \beta_j \)
christoffel	The 3D Christoffel symbols \( \Gamma^k_{ij} \)
alpha	The lapse function \( \alpha \)

Returns

The extrinsic curvature tensor \( K_{ij} \)

References alpha, and beta.

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

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