Provides methods to compute the conformal Ricci tensor \( \tilde{R}_{ij} \) in the BSSN formulation. More...

#include <BSSNRicciTildeTensor.hpp>

Collaboration diagram for tensorium_RG::RicciTildeTensor< T >:

Static Public Member Functions
static tensorium::Tensor< T, 2 >	compute_laplacian_term (const tensorium::Vector< T > &X, T dx, T dy, T dz, const Metric< T > &metric, const tensorium::Tensor< T, 2 > &gamma_tilde_inv)
	Compute \( R^{(1)}_{ij} = -\frac{1}{2} \tilde{\gamma}^{kl} \partial_k \partial_l \tilde{\gamma}_{ij} \) using scalar autodiff.

static tensorium::Tensor< T, 2 >	compute_dGamma_term (const tensorium::Vector< T > &X, T dx, T dy, T dz, const tensorium::Vector< T > &tilde_Gamma, const tensorium::Tensor< T, 2 > &tilde_gamma_contract)
	Compute \( R^{(2)}_{ij} = \frac{1}{2} \left( \nabla_i \tilde{\Gamma}_j + \nabla_j \tilde{\Gamma}_i \right) \).

static tensorium::Tensor< T, 2 >	compute_GammaGamma_term (const tensorium::Vector< T > &tilde_Gamma, const tensorium::Tensor< T, 3 > &christoffel_tilde, const tensorium::Tensor< T, 2 > &gamma_tilde)
	Compute the nonlinear contracted term:

static tensorium::Tensor< T, 2 >	compute_GammaProduct_term (const tensorium::Tensor< T, 2 > &gamma_tilde_inv, const tensorium::Tensor< T, 3 > &christoffel_tilde)
	Compute the quadratic Christoffel term:

static tensorium::Tensor< T, 2 >	compute_Ricci_Tilde_tensor (const ChiContext< T > &chi_context, const tensorium::Tensor< T, 2 > &gamma_tilde_inv, const tensorium::Vector< T > &tilde_Gamma, const tensorium::Tensor< T, 3 > &christoffel_tilde, const tensorium::Tensor< T, 2 > &gamma_tilde)
	Combine all four contributions to compute the conformal Ricci tensor:

Detailed Description

template<typename T>
class tensorium_RG::RicciTildeTensor< T >

Provides methods to compute the conformal Ricci tensor \( \tilde{R}_{ij} \) in the BSSN formulation.

The conformal Ricci tensor is given by:

\[ \tilde{R}_{ij} = R^{(1)}_{ij} + R^{(2)}_{ij} + R^{(3)}_{ij} + R^{(4)}_{ij} \]

where:

\( R^{(1)}_{ij} = -\frac{1}{2} \tilde{\gamma}^{kl} \partial_k \partial_l \tilde{\gamma}_{ij} \)
\( R^{(2)}_{ij} = \frac{1}{2} \left( \nabla_i \tilde{\Gamma}_j + \nabla_j \tilde{\Gamma}_i \right) \)
\( R^{(3)}_{ij} = \frac{1}{2} \tilde{\Gamma}^k \left( \tilde{\gamma}_{ki} \tilde{\Gamma}^l_{jl} + \tilde{\gamma}_{kj} \tilde{\Gamma}^l_{il} \right) \)
\( R^{(4)}_{ij} = \tilde{\gamma}^{\ell m} \left( 2 \tilde{\Gamma}^k_{\ell(i} \tilde{\Gamma}_{j)km} + \tilde{\Gamma}^k_{im} \tilde{\Gamma}_{k\ell j} \right) \)

Each term is computed separately and combined to give \( \tilde{R}_{ij} \).

Member Function Documentation

◆ compute_dGamma_term()

template<typename T >

static tensorium::Tensor< T, 2 > tensorium_RG::RicciTildeTensor< T >::compute_dGamma_term	(	const tensorium::Vector< T > &	X,
		T	dx,
		T	dy,
		T	dz,
		const tensorium::Vector< T > &	tilde_Gamma,
		const tensorium::Tensor< T, 2 > &	tilde_gamma_contract )

inlinestatic

Compute \( R^{(2)}_{ij} = \frac{1}{2} \left( \nabla_i \tilde{\Gamma}_j + \nabla_j \tilde{\Gamma}_i \right) \).

Parameters

X	Spatial point
dx,dy,dz	Grid spacings
tilde_Gamma	Contracted conformal Christoffel vector \( \tilde{\Gamma}^i \)
tilde_gamma	Conformal metric \( \tilde{\gamma}_{ij} \)

Returns: Symmetrized derivative term \( R^{(2)}_{ij} \)

References tensorium_RG::partial_scalar(), and X().

Referenced by tensorium_RG::RicciTildeTensor< T >::compute_Ricci_Tilde_tensor().

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◆ compute_GammaGamma_term()

template<typename T >

static tensorium::Tensor< T, 2 > tensorium_RG::RicciTildeTensor< T >::compute_GammaGamma_term	(	const tensorium::Vector< T > &	tilde_Gamma,
		const tensorium::Tensor< T, 3 > &	christoffel_tilde,
		const tensorium::Tensor< T, 2 > &	gamma_tilde )

inlinestatic

Compute the nonlinear contracted term:

\[ R^{(3)}_{ij} = \frac{1}{2} \tilde{\Gamma}^k \left( \tilde{\gamma}_{ki} \tilde{\Gamma}^l_{jl} + \tilde{\gamma}_{kj} \tilde{\Gamma}^l_{il} \right) \]

Parameters

tilde_Gamma	Contracted Christoffel symbols \( \tilde{\Gamma}^k \)
christoffel_tilde	Christoffel symbols \( \tilde{\Gamma}^i_{jk} \)
gamma_tilde	Conformal metric \( \tilde{\gamma}_{ij} \)

Returns: Tensor \( R^{(3)}_{ij} \)

Referenced by tensorium_RG::RicciTildeTensor< T >::compute_Ricci_Tilde_tensor().

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◆ compute_GammaProduct_term()

template<typename T >

static tensorium::Tensor< T, 2 > tensorium_RG::RicciTildeTensor< T >::compute_GammaProduct_term	(	const tensorium::Tensor< T, 2 > &	gamma_tilde_inv,
		const tensorium::Tensor< T, 3 > &	christoffel_tilde )

inlinestatic

Compute the quadratic Christoffel term:

\[ R^{(4)}_{ij} = \tilde{\gamma}^{\ell m} \left( 2 \tilde{\Gamma}^k_{\ell(i} \tilde{\Gamma}_{j)km} + \tilde{\Gamma}^k_{im} \tilde{\Gamma}_{k\ell j} \right) \]

Parameters

gamma_tilde_inv	Inverse conformal metric \( \tilde{\gamma}^{\ell m} \)
christoffel_tilde	Christoffel symbols \( \tilde{\Gamma}^i_{jk} \)

Returns: Tensor \( R^{(4)}_{ij} \)

Referenced by tensorium_RG::RicciTildeTensor< T >::compute_Ricci_Tilde_tensor().

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◆ compute_laplacian_term()

template<typename T >

static tensorium::Tensor< T, 2 > tensorium_RG::RicciTildeTensor< T >::compute_laplacian_term	(	const tensorium::Vector< T > &	X,
		T	dx,
		T	dy,
		T	dz,
		const Metric< T > &	metric,
		const tensorium::Tensor< T, 2 > &	gamma_tilde_inv )

inlinestatic

Compute \( R^{(1)}_{ij} = -\frac{1}{2} \tilde{\gamma}^{kl} \partial_k \partial_l \tilde{\gamma}_{ij} \) using scalar autodiff.

Parameters

X	Position vector
dx,dy,dz	Grid spacings
metric	Metric object to evaluate \( \tilde{\gamma}_{ij} \)
gamma_tilde_inv	Inverse of the conformal metric \( \tilde{\gamma}^{kl} \)

Returns: Laplacian term \( R^{(1)}_{ij} \)

References tensorium_RG::Metric< T >::BSSN(), chi, and X().

Referenced by tensorium_RG::RicciTildeTensor< T >::compute_Ricci_Tilde_tensor().

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◆ compute_Ricci_Tilde_tensor()

template<typename T >

static tensorium::Tensor< T, 2 > tensorium_RG::RicciTildeTensor< T >::compute_Ricci_Tilde_tensor	(	const ChiContext< T > &	chi_context,
		const tensorium::Tensor< T, 2 > &	gamma_tilde_inv,
		const tensorium::Vector< T > &	tilde_Gamma,
		const tensorium::Tensor< T, 3 > &	christoffel_tilde,
		const tensorium::Tensor< T, 2 > &	gamma_tilde )

inlinestatic

Combine all four contributions to compute the conformal Ricci tensor:

\[ \tilde{R}_{ij} = R^{(1)}_{ij} + R^{(2)}_{ij} + R^{(3)}_{ij} + R^{(4)}_{ij} \]

Parameters

chi_context	ChiContext (holds metric, position, grid spacing, etc.)
gamma_tilde_inv	Inverse conformal metric
tilde_Gamma	Contracted conformal Christoffel vector
christoffel_tilde	Conformal Christoffel symbols
gamma_tilde	Conformal metric

Returns: Tensor \( \tilde{R}_{ij} \)

References tensorium_RG::RicciTildeTensor< T >::compute_dGamma_term(), tensorium_RG::RicciTildeTensor< T >::compute_GammaGamma_term(), tensorium_RG::RicciTildeTensor< T >::compute_GammaProduct_term(), tensorium_RG::RicciTildeTensor< T >::compute_laplacian_term(), and X().

Referenced by tensorium_RG::RicciPhysicalTensor< T >::compute_Ricci_total(), and 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/BSSNRicciTildeTensor.hpp

Static Public Member Functions

Detailed Description

Member Function Documentation

◆ compute_dGamma_term()

◆ compute_GammaGamma_term()

◆ compute_GammaProduct_term()

◆ compute_laplacian_term()

◆ compute_Ricci_Tilde_tensor()