Gravica

General Relativity computation library built on Symbolica (Rust-powered CAS).

Gravica computes the full GR tensor chain — from metric tensor to Einstein tensor — using Symbolica's high-performance symbolic algebra engine, achieving 23x–4300x speedup over EinsteinPy/SymPy.

Features

• Full GR computation chain: Metric → Christoffel → Riemann → Ricci → Einstein → Weyl
• Curvature invariants: Kretschner scalar, Ricci scalar
• Schouten tensor, Stress-Energy-Momentum tensor
• Geodesic equation generator
• Index raising/lowering utilities
• Built-in metrics: Minkowski, Schwarzschild, Kerr, FLRW, Reissner-Nordström, de Sitter, anti-de Sitter, Gödel
• Lazy evaluation with caching
• Cross-validated against EinsteinPy

Documentation

API reference: https://site.jijinbei.jp/gravica/

Quick Start

Tutorial	Topics
Getting Started	Schwarzschild metric, full pipeline (Metric → Christoffel → Riemann → Ricci → Einstein)
Predefined Metrics	All 8 built-in metrics: Minkowski, Schwarzschild, Kerr, FLRW, Gödel, Reissner–Nordström, de Sitter, Anti-de Sitter
Weyl Tensor & Kretschner Scalar	Curvature invariants, singularity detection, vacuum identity $C_{abcd} = R_{abcd}$
Geodesic Equations	Symbolic equations of motion for Schwarzschild and Kerr spacetimes
Index Manipulation	Raising and lowering tensor indices, round-trip verification
Kerr Black Hole	Full tensor pipeline on the Kerr metric, vacuum solution verification
FLRW Cosmology	Stress-energy tensor, cosmological constant, Schouten tensor

Benchmarks: Gravica vs EinsteinPy

All benchmarks measured on the same machine. Median of 3 runs with GC disabled.

Speedup Heatmap

Absolute Time Comparison

Summary

Computation	Minkowski	Schwarzschild	FLRW
Christoffel	33x	23x	23x
Riemann	91x	149x	147x
Ricci	72x	40x	296x
Ricci Scalar	472x	2021x	544x
Einstein	1391x	4342x	1029x

Metric inverse is ~0.3–0.5x (Python cofactor overhead), but this is amortized by the massive speedups in downstream computations.

Reproduce

uv run benchmarks/run_benchmarks.py    # Run benchmarks
uv run benchmarks/plot_benchmarks.py   # Generate charts

Architecture

MetricTensor → ChristoffelSymbols → RiemannTensor → RicciTensor → EinsteinTensor
                      ↓                   ↓              ↓       → WeylTensor
               GeodesicEquations   KretschnerScalar  SchoutenTensor
                                                              ↓
                                                     StressEnergyTensor

Module	Computes
metric.py	$g_{ab}$, $g^{ab}$, $\det(g)$
christoffel.py	$\Gamma^a_{\ bc} = g^{ad},\tfrac{1}{2}(\partial_b,g_{ac} + \partial_c,g_{ab} - \partial_a,g_{bc})$
riemann.py	$R^a_{\ bcd}$, $R_{abcd}$, $R^{abcd}$
ricci.py	$R_{ab} = R^c_{\ acb}$, $R = g^{ab},R_{ab}$
einstein.py	$G_{ab} = R_{ab} - \tfrac{1}{2},g_{ab},R$
weyl.py	$C_{abcd}$ (Weyl conformal tensor)
kretschner.py	$K = R_{abcd},R^{abcd}$ (Kretschner scalar)
geodesic.py	$\ddot{x}^a + \Gamma^a_{\ bc},\dot{x}^b,\dot{x}^c = 0$
schouten.py	$S_{ab} = \tfrac{1}{n-2}\bigl(R_{ab} - \tfrac{R,g_{ab}}{2(n-1)}\bigr)$
stress_energy.py	$8\pi G,T_{ab} = G_{ab} + \Lambda,g_{ab}$
indexing.py	Index raising / lowering for rank-2 tensors

Tests

uv run pytest

Verified properties:

• Minkowski: All tensors $= 0$
• Schwarzschild: $R_{ab} = 0$, $G_{ab} = 0$ (vacuum), $K = 12,r_s^2/r^6$
• Riemann symmetries: $R^a_{\ bcd} = -R^a_{\ bdc}$
• Christoffel known values: $\Gamma^r_{\ tt} = r_s(r-r_s)/(2r^3)$
• de Sitter / anti-de Sitter: Ricci scalar matches analytic values
• Geodesic equations: Free particle in Minkowski
• Index roundtrip: Raise then lower recovers original tensor
• EinsteinPy cross-validation: Christoffel and Ricci match

License

MIT