tensorium_RG Namespace Reference

Classes
class	BSSN
	Driver class to initialize and store BSSN variables from an input spacetime metric. More...
class	BSSNAtildeTensor
	Computes the trace-free conformal extrinsic curvature tensor \( \tilde{A}_{ij} \) in the BSSN formalism. More...
class	BSSNChristoffel
	Compute the conformal Christoffel symbols \( \tilde{\Gamma}^k_{ij} \). More...
struct	BSSNGrid
	Storage structure for all evolved BSSN variables on a single grid point or patch. More...
class	ChristoffelSym
	Stores and computes Christoffel symbols \( \Gamma^\lambda_{\mu\nu} \). More...
class	ExtrinsicCurvature
	Computes the extrinsic curvature tensor \( K_{ij} \) from BSSN variables. More...
class	Metric
	A callable 4D metric class for general relativity (Minkowski, Schwarzschild, Kerr, etc.) More...
class	RicciTensor
	Computes the Ricci tensor and Ricci scalar from a 4D Riemann tensor. More...
class	RiemannTensor
	Computes the 4D Riemann curvature tensor \( R^\rho_{\ \sigma\mu\nu} \). More...
class	TildeGamma
	Computes the contracted conformal Christoffel vector \( \tilde{\Gamma}^i \). More...

Functions
template<typename T >
void	compute_partial_derivatives_3D (const tensorium::Tensor< T, 5 > &gamma_field, size_t i, size_t j, size_t k, T dx, T dy, T dz, tensorium::Tensor< T, 3 > &dgamma_out)
	Compute the 3D partial derivatives \( \partial_k \tilde{\gamma}_{ij} \) from a 5D field tensor.
template<typename T , typename TensorFunc >
void	compute_partial_derivatives_tensor2D (const tensorium::Vector< T > &X, T dx, T dy, T dz, TensorFunc &&func, tensorium::Tensor< T, 3 > &out)
template<typename T , typename VectorFunc >
void	compute_partial_derivatives_vector (const tensorium::Vector< T > &X, T dx, T dy, T dz, VectorFunc &&func, tensorium::Tensor< T, 2 > &out)
template<typename T >
tensorium::Tensor< T, 5 >	generate_conformal_metric_field (const tensorium_RG::Metric< T > &metric, size_t Nx, size_t Ny, size_t Nz, T dx, T dy, T dz)
	Generate the conformal 3-metric field \( \tilde{\gamma}_{ij}(x^i) \) on a 3D grid.
template<typename T >
	__attribute__ ((always_inline, hot, flatten)) inline tensorium
	Compute the inverse of a 2D metric tensor using local matrix inversion.

Function Documentation

__attribute__()

template<typename T >

tensorium_RG::__attribute__

(

(always_inline, hot, flatten)

)

Compute the inverse of a 2D metric tensor using local matrix inversion.

Compute Christoffel symbols at an offset position along one direction.

Converts the tensor to a matrix, computes the inverse, and converts back.

Template Parameters

T	Scalar type

Parameters

g	Metric tensor \( g_{\mu\nu} \)

Returns

Inverse tensor \( g^{\mu\nu} \)

Used in numerical relativity to evaluate derivatives of Christoffel symbols.

Template Parameters

T	Scalar type

Parameters

X	Base coordinate
direction	Axis of offset
offset	Value to add (positive or negative)
h	Finite difference step size
g	Output metric tensor
g_inv	Output inverse metric
Gamma_out	Output Christoffel tensor
metric	Metric object

References tensorium::Tensor< K, Rank >::dimensions.

compute_partial_derivatives_3D()

template<typename T >

void tensorium_RG::compute_partial_derivatives_3D ( const tensorium::Tensor< T, 5 > & gamma_field, size_t i, size_t j, size_t k, T dx, T dy, T dz, tensorium::Tensor< T, 3 > & dgamma_out )

inline

Compute the 3D partial derivatives \( \partial_k \tilde{\gamma}_{ij} \) from a 5D field tensor.

Uses centered 4th-order or 2nd-order finite differences for inner and boundary points.

\[ \partial_k \tilde{\gamma}_{ij} = \begin{cases} \frac{-f(x+2h) + 8f(x+h) - 8f(x-h) + f(x-2h)}{12h} & \text{if inner} \\ \frac{f(x+h) - f(x-h)}{2h} & \text{if near boundary} \\ 0 & \text{otherwise} \end{cases} \]

Parameters

gamma_field	Field of type Tensor<T, 5> with dimensions [Nx, Ny, Nz, 3, 3]
i,j,k	Grid position
dx,dy,dz	Grid spacings
dgamma_out	Output tensor \( \partial_k \tilde{\gamma}_{ij} \)

References tensorium::Tensor< K, Rank >::resize(), and tensorium::Tensor< K, Rank >::shape().

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compute_partial_derivatives_tensor2D()

template<typename T , typename TensorFunc >

void tensorium_RG::compute_partial_derivatives_tensor2D	(	const tensorium::Vector< T > &	X,
		T	dx,
		T	dy,
		T	dz,
		TensorFunc &&	func,
		tensorium::Tensor< T, 3 > &	out )

References tensorium::Tensor< K, Rank >::resize(), and X().

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compute_partial_derivatives_vector()

template<typename T , typename VectorFunc >

void tensorium_RG::compute_partial_derivatives_vector	(	const tensorium::Vector< T > &	X,
		T	dx,
		T	dy,
		T	dz,
		VectorFunc &&	func,
		tensorium::Tensor< T, 2 > &	out )

References tensorium::Tensor< K, Rank >::resize(), and X().

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generate_conformal_metric_field()

template<typename T >

tensorium::Tensor< T, 5 > tensorium_RG::generate_conformal_metric_field	(	const tensorium_RG::Metric< T > &	metric,
		size_t	Nx,
		size_t	Ny,
		size_t	Nz,
		T	dx,
		T	dy,
		T	dz )

Generate the conformal 3-metric field \( \tilde{\gamma}_{ij}(x^i) \) on a 3D grid.

For each spatial grid point \( (x, y, z) \), the spatial metric \( \gamma_{ij} \) is extracted from the given metric, the conformal factor \( \chi \) is computed, and the conformal metric is defined as:

\[ \tilde{\gamma}_{ij} = \chi \, \gamma_{ij} \]

Template Parameters

T	Scalar type (typically double)

Parameters

metric	A metric object providing BSSN(X, α, β, γ) and compute_conformal_factor(γ)
Nx,Ny,Nz	Grid resolution in x, y, z directions
dx,dy,dz	Grid spacings in x, y, z directions

Returns

A 5D tensor with shape \( (N_x, N_y, N_z, 3, 3) \) containing \( \tilde{\gamma}_{ij} \) at each grid point

< Coordinates X^μ = (t, x, y, z)

< Time is fixed at t = 0

References alpha, beta, tensorium_RG::Metric< T >::BSSN(), chi, tensorium_RG::Metric< T >::compute_conformal_factor(), tensorium_RG::Metric< T >::compute_conformal_metric(), gamma, and X().

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