tensorium_RG::RicciPhysicalTensor< T > Class Template Reference

Computes the physical 3-Ricci tensor \( R_{ij} \) as the sum of conformal and conformal-factor contributions. More...

#include <BSSNRicciComplete.hpp>

Collaboration diagram for tensorium_RG::RicciPhysicalTensor< T >:

Static Public Member Functions
static tensorium::Tensor< T, 2 >	compute_Ricci_total (const ChiContext< T > &chi_context, const tensorium::Tensor< T, 2 > &gamma_tilde, const tensorium::Tensor< T, 2 > &gamma_tilde_inv, const tensorium::Vector< T > &tilde_Gamma, const tensorium::Tensor< T, 3 > &christoffel_tilde)
	Compute the full Ricci tensor \( R_{ij} \) as the sum of \( \tilde{R}_{ij} \) and \( R^\chi_{ij} \).

Detailed Description

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

Computes the physical 3-Ricci tensor \( R_{ij} \) as the sum of conformal and conformal-factor contributions.

In the BSSN formulation of general relativity, the physical Ricci tensor \( R_{ij} \) is decomposed as:

\[ R_{ij} = \tilde{R}_{ij} + R^{\chi}_{ij} \]

where:

• \( \tilde{R}_{ij} \) is the Ricci tensor of the conformal metric \( \tilde{\gamma}_{ij} \)
• \( R^{\chi}_{ij} \) encodes the contributions from the conformal factor \( \chi \)

Expanding all terms in partial derivatives:

\[ R_{ij} = -\frac{1}{2} \tilde{\gamma}^{kl} \partial_k \partial_l \tilde{\gamma}_{ij} + \frac{1}{2} \left( \partial_i \tilde{\Gamma}_j + \partial_j \tilde{\Gamma}_i \right) + \frac{1}{2} \tilde{\Gamma}^k \left( \tilde{\gamma}_{ki} \tilde{\Gamma}^l_{jl} + \tilde{\gamma}_{kj} \tilde{\Gamma}^l_{il} \right) + \tilde{\gamma}^{\ell m} \left( 2 \tilde{\Gamma}^k_{\ell(i} \tilde{\Gamma}_{j)km} + \tilde{\Gamma}^k_{im} \tilde{\Gamma}_{k\ell j} \right) + \frac{1}{\chi} \left( \partial_i \partial_j \chi - \tilde{\Gamma}^k_{ij} \partial_k \chi \right) + \frac{1}{2\chi} \tilde{\gamma}_{ij} \tilde{\gamma}^{kl} \left( \partial_k \partial_l \chi - \tilde{\Gamma}^m_{kl} \partial_m \chi \right) - \frac{3}{2\chi^2} \partial_i \chi \partial_j \chi + \frac{1}{2\chi^2} \tilde{\gamma}_{ij} \tilde{\gamma}^{kl} \partial_k \chi \partial_l \chi \]

This class computes each term independently and sums them to obtain the total physical Ricci tensor.

Template Parameters

T	Scalar type (e.g., float or double)

Member Function Documentation

compute_Ricci_total()

template<typename T >

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

inlinestatic

Compute the full Ricci tensor \( R_{ij} \) as the sum of \( \tilde{R}_{ij} \) and \( R^\chi_{ij} \).

This method calls:

• RicciTildeTensor::compute_Ricci_Tilde_tensor()
• RicciConformalTensor::compute_Ricci_chi_total()

and returns:

\[ R_{ij} = \tilde{R}_{ij} + R^{\chi}_{ij} \]

Also prints the Frobenius norm of \( R_{ij} \) for diagnostics.

Parameters

chi_context	ChiContext object containing metric, grid spacing, and position
gamma_tilde	Conformal metric \( \tilde{\gamma}_{ij} \)
gamma_tilde_inv	Inverse conformal metric \( \tilde{\gamma}^{ij} \)
tilde_Gamma	Contracted conformal Christoffel symbols \( \tilde{\Gamma}^i \)
christoffel_tilde	Full Christoffel symbols \( \tilde{\Gamma}^i_{jk} \)

Returns

Physical Ricci tensor \( R_{ij} \)

References tensorium_RG::RicciConformalTensor< T >::compute_Ricci_chi_total(), and tensorium_RG::RicciTildeTensor< T >::compute_Ricci_Tilde_tensor().

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/BSSNRicciComplete.hpp