Strong-Field Geodesic Study with EinsteinPy
A completed computational physics study of timelike and null geodesics in Schwarzschild and Kerr spacetime using EinsteinPy and independent analytic benchmarks.
Software compatibility
The released source, retained results, tests and documents use Python 3.12 and EinsteinPy 0.4.0. Mathematica, MATLAB and proprietary relativity software files are not included.
Project definition
Problem statement
A visually plausible relativistic trajectory is not enough to establish numerical accuracy. Initial conditions, coordinate conventions, approximation limits and conserved quantities must support the same physical interpretation.
The physics problem is to compare EinsteinPy trajectories with independent Schwarzschild and Kerr references while retaining enough evidence to identify strong-field approximation failure and integration error.
Project objectives
- Calculate equatorial circular orbit radii, constants and coordinate frequencies for selected Kerr spins.
- Measure eccentric Schwarzschild periapsis advance and compare it with exact quadrature and the leading post-Newtonian expression.
- Calculate exact null deflection and turning radii above the critical impact parameter.
- Track Hamiltonian, energy, angular momentum and radial drift for the integrated trajectories.
- Compare three integration orders across three step sizes.
Project structure
Project components
Analytic physics
Calculates horizons, critical radii, circular constants, exact quadrature references and weak-field approximations.
EinsteinPy integration
Builds and runs timelike and null Schwarzschild and Kerr trajectories with declared covariant momenta.
Diagnostics
Measures Hamiltonian normalization, drift, extrema, frequencies, precession and turning-radius agreement.
Experiment workflow
Runs the circular, eccentric, null and convergence matrices with deterministic configuration.
Evidence
Retains CSV, JSON, compressed trajectories, figures, tests, references and complete documentation.
Methodology
Project workflow
- 01Load the matrix
A fixed configuration declares spins, orbit sizes, eccentricities, impact parameters, orders and step sizes.
- 02Build initial states
Independent formulas define the expected constants, turning points and physical branch.
- 03Integrate trajectories
EinsteinPy advances the selected timelike and null geodesics.
- 04Compare evidence
The workflow calculates analytic errors, invariant residuals and convergence trends.
- 05Verify the release
Tests, coverage, dependency audit, package build and document checks confirm the retained delivery.
Demonstration scenario
The student rebuilds all 32 cases, shows circular frequency agreement below 4.54e-14 relative error, compares exact and EinsteinPy eccentric precession, then explains why weak-field precession and deflection formulas become inaccurate close to the compact object.
Engineering
Tools and method
- Tools
- The project uses EinsteinPy 0.4.0, Python 3.12, NumPy, SciPy, Pandas, Matplotlib for subject analysis, simulation, and results.
- Relativity engine
- EinsteinPy 0.4.0 provides the Hamiltonian geodesic integrator for Schwarzschild and Kerr metrics.
- Independent references
- SciPy quadrature and closed formulas provide separate frequency, precession, deflection and turning-point benchmarks.
- Evidence analysis
- NumPy and Pandas retain the complete case matrices and trajectory diagnostics.
- Figures
- Matplotlib generates twelve labelled figures directly from the released results.
- Reproducibility
- Pinned dependencies, exact configuration, automated tests, raw evidence, Word files and PDFs support independent reruns.
Testing
Evaluation
Evaluation measures
- Circular Kerr frequency error and radial drift
- Exact and EinsteinPy eccentric periapsis advance
- Leading post-Newtonian approximation error
- Exact and weak-field null deflection
- Integrated null turning-radius error
- Hamiltonian, energy and angular-momentum drift
- Integrator order and step-size convergence
Project boundaries
- The study uses ideal Schwarzschild and Kerr vacuum metrics in geometrized units.
- The retained eccentric and null trajectories are equatorial and do not cover the full geodesic parameter space.
- Matter, plasma, radiation reaction, self-force, spin-curvature coupling and metric perturbations are outside the model.
- The project is a numerical verification study, not an astronomical fit, black-hole image or experimental test of gravity.
- The physical claims apply only to the released case matrix and declared numerical settings.
Included
- 01Complete Python and EinsteinPy source code
- 02Twelve circular Schwarzschild and Kerr cases
- 03Four eccentric precession cases
- 04Seven null-deflection cases and three integrated null trajectories
- 05Nine integrator-convergence cases
- 06Four retained result tables and three compressed trajectory datasets
- 07Twelve labelled project figures and two sourced literature images
- 08Fifty-six automated tests with 99.01 percent package coverage
- 09Complete project files, calculations, results and analysis material in a private GitHub repository
- 1075-page project documentation in PDF and editable Word formats
- 1116-page setup and usage guide in PDF and editable Word formats
- 12Sixty annotated references
Project record
No information is collected on this page.
- Permanent project ID
- GP-PH-1MYEVO1
- Catalogued
- 21 Aug 2026
- Completed
- 29 Aug 2026
- Verified
- 29 Aug 2026
- Demonstration
- Included in repository
Handover
After purchase
- 01Payment is confirmed
The project is marked unavailable and cannot be purchased again.
- 02Repository access is granted
The buyer's submitted GitHub account receives access to the private repository.
- 03The purchase record is delivered
The certification sheet is prepared from the reviewed buyer details and sent privately by email.