Tensorium — DiffGeometry Module

This directory provides the core components for differential geometry computations in general relativity. It supports both symbolic 4D analytical calculations and the foundations for ADM/BSSN-based 3+1 numerical simulations.

Purpose

This module enables:

• Representation and manipulation of the spacetime metric
• Computation of Christoffel symbols (first and second kind)
• Construction of curvature tensors: Riemann, Ricci, and Ricci scalar
• (Upcoming) Support for the BSSN conformal formalism on spatial grids

Structure

DiffGeometry/
├── Metric.hpp              // Spacetime metric representation
├── ChristoffelSymbol.hpp   // Computation of Γ^k_{ij}
├── RicciTensor.hpp         // Ricci tensor computation
├── RiemannTensor.hpp       // Full Riemann tensor computation
├── Tensor.hpp              // Basic tensor structures
└── BSSN/                   // Conformal BSSN variables and evolution (WIP)

Features

• Metric provides the metric components, inverse metric, and determinant.
• ChristoffelSymbol computes the Christoffel symbols from metric derivatives.
• RicciTensor constructs the Ricci tensor by contracting the Riemann tensor.
• RiemannTensor provides full access to the Riemann curvature tensor ( R^\mu_{\ \nu\rho\sigma} ).
• BSSN/ (to be implemented) will handle the conformal metric decomposition, trace-free extrinsic curvature, and conformal connection functions.

Internal Dependencies

• Uses Tensor.hpp from Core/ for generic tensor representation.
• Uses DerivateND from Core/ for partial derivatives (via centered difference or spectral methods).
• Fully self-contained, with no external library dependency.

Current Supported Metrics

• Minkowski (flat spacetime)
• Schwarzschild (non-rotating black hole)
• Kerr (rotating black hole)

Example: Compute Riemann and Ricci Tensors from Kerr Metric

constexpr size_t dim = 4;
 
tensorium::Vector<double> X(dim);
X(0) = 0.0;
X(1) = 10.0;
X(2) = M_PI / 2.0;
X(3) = 0.0;
 
tensorium::Tensor<double, 2> g({dim, dim});
tensorium::Tensor<double, 2> g_inv({dim, dim});
 
tensorium_RG::Metric<double> metric("kerr", 1.0, 0.8);
metric(X, g);
 
g_inv = tensorium::inv_mat_tensor(g);
 
auto gamma = tensorium::compute_christoffel(X, 1e-5, g, g_inv, metric);
auto Riemann = tensorium::compute_riemann_tensor<double>(X, 1e-5, metric);
auto Ricci = tensorium::contract_tensor<0, 2>(Riemann); // Ricci^μ_μ = R_{ρσ}

Status

The module is stable for 4D analytical computations. Symbolic initializations and automation through the Symbolics/ and BSSN modules are under active development.

References

• M. Alcubierre, Introduction to 3+1 Numerical Relativity
• T. W. Baumgarte & S. L. Shapiro, Numerical Relativity: Solving Einstein's Equations on the Computer