srfm::tensor::ChristoffelSymbols Class Reference

#include <tensor.hpp>

Public Member Functions
	ChristoffelSymbols (const MetricTensor &metric, double h=constants::DEFAULT_FD_STEP)
ChristoffelArray	compute (const SpacetimePoint &x) const
FourVelocity	contract (const ChristoffelArray &gamma, const FourVelocity &u) const

Detailed Description

Computes the Christoffel symbols of the second kind Γ^λ_μν at a spacetime point by numerically differentiating the metric tensor.

The Christoffel symbols measure how the metric — and therefore the market covariance structure — changes from one market state to another. They are the "connection" that converts coordinate changes into physical changes in the correlation geometry.

Formula (Einstein summation): Γ^λ_μν = ½ g^λσ (∂_μ g_νσ + ∂_ν g_μσ − ∂_σ g_μν)

Partial derivatives are computed via central finite differences: ∂g_μν/∂x^σ ≈ [g_μν(x + h·ê_σ) − g_μν(x − h·ê_σ)] / (2h)

Definition at line 356 of file tensor.hpp.

Constructor & Destructor Documentation

ChristoffelSymbols()

srfm::tensor::ChristoffelSymbols::ChristoffelSymbols ( const MetricTensor &  metric, double  h = constants::DEFAULT_FD_STEP  )

explicit

Construct from a metric tensor.

Arguments

• metric — The position-dependent metric (held by const-reference)
• h — Finite-difference step for metric derivatives (default 1e-5)

Definition at line 14 of file christoffel.cpp.

Member Function Documentation

compute()

ChristoffelArray srfm::tensor::ChristoffelSymbols::compute

(

const SpacetimePoint &

x

)

const

Compute all 4³ = 64 Christoffel symbols at point x.

Arguments

• x — Spacetime point at which to evaluate Γ^λ_μν

Returns

Array indexed as result[lambda](mu, nu) = Γ^λ_μν. Returns all-zero array if the metric is singular at x.

Definition at line 33 of file christoffel.cpp.

contract()

FourVelocity srfm::tensor::ChristoffelSymbols::contract	(	const ChristoffelArray &	gamma,
		const FourVelocity &	u
	)		const

Contract the Christoffel symbols with a four-velocity: result^λ = Γ^λ_μν u^μ u^ν

This is the RHS of the geodesic acceleration equation (negated).

Definition at line 79 of file christoffel.cpp.

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

• include/srfm/tensor.hpp
• src/tensor/christoffel.cpp