tensorium_RG::RicciTensor< T > Class Template Reference

Computes the Ricci tensor and Ricci scalar from a 4D Riemann tensor. More...

#include <RicciTensor.hpp>

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Public Types
using	Tensor4D = tensorium::Tensor<T, 4>
using	Tensor2D = tensorium::Tensor<T, 2>
using	VectorT = tensorium::Vector<T>

Static Public Member Functions
static tensorium::Tensor< T, 2 >	contract_to_ricci (const Tensor4D &Riemann, const tensorium::Tensor< T, 2 > &g_inv)
	Contracts a 4D Riemann tensor into the 2D Ricci tensor.
static double	compute_ricci_scalar (const Tensor2D &Ricci, const tensorium::Tensor< T, 2 > &g_inv)
	Computes the Ricci scalar curvature.
static void	print_componentwise (const Tensor2D &R, T threshold=1e-12)
	Prints the Ricci tensor components with optional zero threshold.
static void	print (const Tensor2D &R, const std::string &name="Ricci")
	Prints only the non-zero components of the Ricci tensor.
static void	print (const Tensor2D &R, const tensorium::Tensor< T, 2 > &g_inv, const std::string &name="Ricci")
	Prints non-zero components using both Ricci and inverse metric.
static void	print_ricci_scalar (const Tensor2D &Ricci, const tensorium::Tensor< T, 2 > &g_inv)
	Computes and prints the Ricci scalar from the tensor and metric.

Detailed Description

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

Computes the Ricci tensor and Ricci scalar from a 4D Riemann tensor.

This class provides static methods to contract the Riemann tensor into the Ricci tensor, compute the scalar curvature, and print these results.

The Ricci tensor is obtained by contraction:

\[ R_{\mu\nu} = R^\rho_{\ \mu\rho\nu} = g^{\rho\sigma} R_{\rho\sigma\mu\nu} \]

The Ricci scalar is the trace:

\[ R = g^{\mu\nu} R_{\mu\nu} \]

Member Typedef Documentation

Tensor2D

template<typename T >

using tensorium_RG::RicciTensor< T >::Tensor2D = tensorium::Tensor<T, 2>

Tensor4D

template<typename T >

using tensorium_RG::RicciTensor< T >::Tensor4D = tensorium::Tensor<T, 4>

VectorT

template<typename T >

using tensorium_RG::RicciTensor< T >::VectorT = tensorium::Vector<T>

Member Function Documentation

compute_ricci_scalar()

template<typename T >

static double tensorium_RG::RicciTensor< T >::compute_ricci_scalar ( const Tensor2D & Ricci, const tensorium::Tensor< T, 2 > & g_inv )

inlinestatic

Computes the Ricci scalar curvature.

Uses the trace formula: \( R = g^{\mu\nu} R_{\mu\nu} \)

Parameters

Ricci	The Ricci tensor \( R_{\mu\nu} \)
g_inv	The inverse metric \( g^{\mu\nu} \)

Returns

The scalar curvature \( R \)

References mu, and nu.

Referenced by tensorium::compute_ricci_scalar(), and tensorium_RG::RicciTensor< T >::print_ricci_scalar().

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contract_to_ricci()

template<typename T >

static tensorium::Tensor< T, 2 > tensorium_RG::RicciTensor< T >::contract_to_ricci ( const Tensor4D & Riemann, const tensorium::Tensor< T, 2 > & g_inv )

inlinestatic

Contracts a 4D Riemann tensor into the 2D Ricci tensor.

Uses the identity \( R_{\mu\nu} = g^{\rho\sigma} R_{\rho\sigma\mu\nu} \)

Parameters

Riemann	The Riemann tensor \( R_{\rho\sigma\mu\nu} \)
g_inv	The inverse metric \( g^{\rho\sigma} \)

Returns

The Ricci tensor \( R_{\mu\nu} \)

References tensorium::Tensor< K, Rank >::fill(), mu, nu, rho, and sigma.

Referenced by tensorium::contract_riemann_to_ricci().

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print() [1/2]

template<typename T >

static void tensorium_RG::RicciTensor< T >::print ( const Tensor2D & R, const std::string & name = "Ricci" )

inlinestatic

Prints only the non-zero components of the Ricci tensor.

Parameters

R	The Ricci tensor
name	Optional tensor name label

References mu, and nu.

print() [2/2]

template<typename T >

static void tensorium_RG::RicciTensor< T >::print ( const Tensor2D & R, const tensorium::Tensor< T, 2 > & g_inv, const std::string & name = "Ricci" )

inlinestatic

Prints non-zero components using both Ricci and inverse metric.

Parameters

R	The Ricci tensor
g_inv	The inverse metric
name	Optional tensor name label

References mu, and nu.

print_componentwise()

template<typename T >

static void tensorium_RG::RicciTensor< T >::print_componentwise ( const Tensor2D & R, T threshold = 1e-12 )

inlinestatic

Prints the Ricci tensor components with optional zero threshold.

Parameters

R	The Ricci tensor
threshold	Minimum value to display (default = 1e-12)

References mu, and nu.

Referenced by tensorium::print_ricci_tensor().

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print_ricci_scalar()

template<typename T >

static void tensorium_RG::RicciTensor< T >::print_ricci_scalar ( const Tensor2D & Ricci, const tensorium::Tensor< T, 2 > & g_inv )

inlinestatic

Computes and prints the Ricci scalar from the tensor and metric.

Parameters

Ricci	The Ricci tensor
g_inv	The inverse metric

References tensorium_RG::RicciTensor< T >::compute_ricci_scalar().

Referenced by tensorium::print_ricci_scalar().

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

• includes/Tensorium/DiffGeometry/RicciTensor.hpp