lyra-geometry 0.1.11

Biblioteca em Python para geometria diferencial simbolica: tensores, conexoes e curvatura.

Lyra Geometry

Lyra Geometry is a Python library for symbolic differential geometry with a focus on tensor spaces, connections, and curvature in Lyra geometry. It is built on SymPy to support exact tensor manipulation in scripts and notebooks.

Highlights

• Tensor spaces with index notation (up/down indices).
• Lyra connection and curvature tensors derived from a metric.
• Automatic Einstein summation for repeated labels.
• Covariant derivatives, torsion, and non-metricity support.
• Friendly API designed for interactive exploration.

Requirements

• Python >= 3.9
• SymPy >= 1.12

Installation

python -m pip install lyra-geometry

For local development:

python -m pip install -e .[dev]

Minimal API reference

Core objects:

• SpaceTime(coords, metric, connection_strategy=...): main high-level API for metrics, connections, and curvature.

  st = pl.SpaceTime(coords=(x, y), metric=sp.diag(1, 1))
  st.riemann
  

• TensorSpace(dim, labels=...): lower-level tensor space without a metric.

  ts = pl.TensorSpace(dim=2, labels=("x", "y"))
  

• Tensor: tensor container with index-aware arithmetic and contractions.

  t = st.tensor.generic("T", (pl.U, pl.D))
  t[+a, -b]
  

• Index, UpIndex, DownIndex: index labels and variance markers.

  a, b = st.index("a b")
  st.g[-a, -b]
  

• U, D: convenience aliases for up/down index variance.

  v = st.tensor.generic("v", (pl.U,))
  w = st.tensor.generic("w", (pl.D,))
  

Connection and curvature strategies:

• LyraConnectionStrategy, LyraCurvatureStrategy: default Lyra geometry behavior.

  st = pl.SpaceTime(coords=(x, y), metric=sp.diag(1, 1))
  st.gamma
  

• FixedConnectionStrategy: use a fixed connection tensor.

  Gamma0 = sp.ImmutableDenseNDimArray([0] * 8, (2, 2, 2))
  st = pl.SpaceTime(coords=(x, y), metric=sp.diag(1, 1),
                    connection_strategy=pl.FixedConnectionStrategy(Gamma0))
  

• Connection, ConnectionTensor, CurvatureStrategy: building blocks for custom strategies.

Helpers:

• example_indexing(): small demo of index notation.

  pl.example_indexing()
  

• greek(): helper for common Greek index labels.

  mu, nu = pl.greek("mu nu")
  

What lyra-geometry is (and what it is not)

What it is

The lyra-geometry is a symbolic library to work with relativistic gravity thoeries. Concretely, lyra-geometry provides:

• Exact symbolic tensor calculus built on top of SymPy.
• A clean, object-oriented SpaceTime abstraction that organizes:
  • metric, inverse metric, and determinant,
  • Levi–Civita connection (by default),
  • Riemann, Ricci, Einstein tensors, and scalar curvature,
  • covariant derivatives with correct index bookkeeping.
• Explicit index variance (up/down indices) with:
  • readable index notation,
  • automatic Einstein summation on repeated labels.
• A workflow designed for interactive notebooks and analytic derivations, where expressions remain transparent and inspectable at every step.

In this default configuration, lyra-geometry behaves exactly as a symbolic GR toolkit.

Where the library extends beyond standard GR is in its native support for Lyra geometry and related non-Riemannian structures:

• the Lyra scale field $\phi$ can be introduced when desired,
• alternative connection strategies,
• optional torsion and non-metricity sectors.

These features are opt-in. They do not interfere with ordinary GR unless explicitly enabled.

In short:
lyra-geometry is a symbolic GR library first, and a Lyra-geometry laboratory second.

---

What it is NOT

To set expectations clearly, lyra-geometry is not:

• A numerical relativity or cosmological simulation framework.
• A high-performance or HPC-oriented tensor engine.
• A “black-box” solver that automatically derives or solves field equations.
• A replacement for large, monolithic CAS environments (e.g. full SageMath).
• A library that enforces Lyra geometry or non-Riemannian assumptions by default.

If you never introduce a scale field, torsion, or non-metricity, the library remains purely Riemannian and fully compatible with standard GR practice.

---

When it makes sense to use it

lyra-geometry is particularly well suited if you want to:

• perform symbolic GR calculations with explicit index control,
• derive connections and curvature tensors from a metric in a reproducible way,
• check lengthy analytic derivations while minimizing index errors,
• work entirely in Python notebooks with a clean, inspectable API,
• explore extensions of GR (Lyra geometry, alternative connections) without rewriting your tensor infrastructure.

If your primary goal is large-scale numerics, data-driven cosmology, or observational pipelines, this library is best used in combination with other tools, or not at all.

Comparison with common GR / tensor libraries

Library	Symbolic	Index Notation	Automatic Contraction	Curvature & Connection	Lyra Geometry	Numeric Focus	Philosophy
SymPy (tensor)	✅	⚠️ (low-level)	⚠️ manual	⚠️ partial	❌	❌	General-purpose CAS
EinsteinPy	✅	✅	⚠️ partial	✅ (GR)	❌	✅	End-to-end GR
SageManifolds	✅	✅	✅	✅ (very rich)	❌	⚠️	Full math environment
Cadabra	✅	✅	✅ (very strong)	⚠️ user-defined	❌	❌	QFT/GR CAS
OGRePy	✅	✅	✅	✅ (GR)	❌	❌	OO GR toolkit
RicciPy	✅	⚠️	⚠️	✅ (GR)	❌	❌	Ricci calculus
Pytearcat	✅	✅	✅	⚠️ limited	❌	❌	GR calculator
galgebra	✅	❌	❌	❌	❌	❌	Geometric algebra
PyMetric	⚠️	❌	❌	⚠️	❌	⚠️	Coord-based DG
lyra-geometry	✅	✅ (core)	✅ (automatic)	✅ (Lyra & Riemann)	✅	❌	Symbolic DG focused on Lyra

Getting started

Create a space with a metric, then inspect its basic objects:

import sympy as sp
import lyra_geometry as pl

x, y = sp.symbols("x y", real=True)
metric = sp.diag(x + 2*y, x**2 * y)

st = pl.SpaceTime(coords=(x, y), metric=metric)

st.g          # metric tensor
st.metric_inv # inverse metric matrix
st.detg       # determinant of the metric

Indices, raising/lowering, and components

Use index labels with explicit variance markers to access components:

a, b, c = st.index("a b c")

st.g[-a, -b]  # g_ab
st.g[+a, +b]  # g^ab

st.g[-a, -b](0, 0)  # component access

Bare indices (like t[a, b]) are not accepted; always use +a/-a.

Generic tensors and contraction

Create symbolic tensors and let repeated labels contract automatically:

v = st.tensor.generic("v", (pl.U,))
w = st.tensor.generic("w", (pl.D,))

scalar = v[+a] * w[-a]  # automatic contraction

Explicit contraction and a simple index-string parser are also available:

st.contract(v[+a], w[-a])
st.eval_contract("v^a w_a")

Covariant derivative

The Lyra covariant derivative adds one covariant index:

dv = st.nabla(v)
dv.signature  # (D, U)
dv[-b, +a](0, 0)

Connection and curvature

When a metric is provided, the Lyra connection and curvature tensors are computed automatically:

st.gamma            # Gamma^a_{bc}
st.riemann          # Riemann tensor
st.ricci            # Ricci tensor
st.einstein         # Einstein tensor
st.scalar_curvature # scalar curvature

Scale, torsion, and non-metricity

You can set a scale field and provide torsion/non-metricity explicitly:

phi = sp.Function("phi")(x)
st.set_scale(phi)

st.set_torsion(st.zeros((pl.D, pl.D, pl.D)))
st.set_nonmetricity(st.zeros((pl.U, pl.D, pl.D)))

st.update()

Custom connection strategies

If you already have Gamma components, you can fix the connection manually:

Gamma0 = sp.ImmutableDenseNDimArray([0] * (2**3), (2, 2, 2))

st2 = pl.SpaceTime(
    coords=(x, y),
    metric=sp.diag(1, 1),
    connection_strategy=pl.FixedConnectionStrategy(Gamma0),
)

st2.gamma

Simplification with fmt

Use fmt() to expand and simplify expressions:

st.ricci.fmt()
st.einstein.fmt()
st.scalar_curvature.fmt()

Notebook examples

The notebook examples/example.ipynb walks through:

• A quick tutorial of the core API.
• Schwarzschild spacetime: metric, connection, and curvature tensors.
• FLRW spacetime: metric, curvature, and covariant derivative of a scalar.
• A spherically symmetric LyST solution with scale field and field equations.

Project structure

• src/lyra_geometry/core.py: core implementation.
• src/lyra_geometry/__init__.py: public exports and version.
• examples/example.ipynb: tutorial and physics examples.
• tests/: pytest smoke tests.

Development and testing

python -m pytest

License

MIT. See LICENSE.