Proper Time Geometry

中文说明 | Bilingual overview

Pedagogical derivations and numerical checks for proper-time rates in special and general relativity, organized around a local energy-momentum angle.

This repository does not propose a new physical theory. It is a teaching and research-note project that repackages standard SR/GR relations in a geometric form that may be easier to inspect, teach, and test.

What This Is

• A geometric parametrization of Lorentz time dilation using an energy-momentum triangle.
• A set of derivations showing how the proper-time rate appears from the invariant interval.
• A static-spacetime decomposition of the form

$$\frac{d\tau}{dt}=G(x)\sin\theta$$

where G(x)=sqrt(g00) is the static metric factor and sin(theta) is the local special-relativistic motion factor.

• Numerical checks for weak-field GPS-scale corrections, Hamilton-Jacobi angle generation, and Schwarzschild local tetrads.
• A public-facing Chinese article draft suitable for popular-science publishing.

What This Is Not

• It is not an alternative to general relativity.
• It does not derive the Einstein field equations.
• It does not currently make observational predictions different from SR/GR.
• It should not be presented as a new gravity theory.

The intended value is conceptual clarity: a compact way to see how energy, momentum, local velocity, background geometry, and proper time fit together.

Core Idea

For a massive particle in special relativity:

$$E^2=p^2c^2+m^2c^4$$

This can be drawn as a right triangle with hypotenuse E and legs pc and mc^2.

Define a local energy-momentum angle:

$$\cos\theta=\frac{pc}{E}, \qquad \sin\theta=\frac{mc^2}{E}$$

Using E=gamma mc^2, one gets:

$$\sin\theta=\frac{1}{\gamma} =\sqrt{1-\frac{v^2}{c^2}}$$

Therefore:

$$\frac{d\tau}{dt}=\sin\theta$$

In a static, zero-shift spacetime,

$$ds^2=c^2d\tau^2=g_{00}(x)c^2dt^2-h_{ij}(x)dx^idx^j$$

the standard line element gives:

$$\frac{d\tau}{dt} = \sqrt{g_{00}(x)} \sqrt{1-\frac{v_{\mathrm{local}}^2}{c^2}}$$

With

$$G(x)=\sqrt{g_{00}(x)}, \qquad \sin\theta=\sqrt{1-\frac{v_{\mathrm{local}}^2}{c^2}},$$

this becomes:

$$\frac{d\tau}{dt}=G(x)\sin\theta$$

This factorization is derived from the standard GR line element in the stated static, zero-shift setting. It is not a universal coordinate formula for arbitrary non-static or shifted metrics.

Notation

The technical notes often write the angle as theta_S, emphasizing its Hamilton-Jacobi/action or phase origin. The popular article writes it as theta_E, emphasizing the energy-momentum triangle. In this project, both refer to the same local energy-momentum angle when used in the SR/GR proper-time context.

Repository Map

File	Purpose
README.md	English project overview.
README.zh-CN.md	Chinese project overview.
BILINGUAL_OVERVIEW.md	Side-by-side bilingual summary.
TECHNICAL_OVERVIEW.zh-CN.md	Original Chinese technical overview and research positioning.
DERIVATION.md	Main derivation notes and boundaries.
GEOMETRIC_ORIGIN.md	Discussion of the geometric origin of G(x) and the angle parameter.
RELATION_TO_GR.md	Relation to standard GR and what is or is not new.
ZHIHU_ARTICLE.md	Chinese popular-science article draft.
derive_predictions.py	Basic numerical checks and generated plots.
experiment_phi_gradient.py	Test A: why a simple potential-gradient direction is not enough.
experiment_hamilton_jacobi.py	Test B: Hamilton-Jacobi/action origin of the angle parameter.
experiment_schwarzschild_tetrad.py	Test C: local tetrad checks in Schwarzschild spacetime.
output/	Generated CSV, JSON, and PNG outputs.

Reproduce the Outputs

Python 3.10 or later is recommended.

With uv:

uv run python derive_predictions.py
uv run python experiment_phi_gradient.py
uv run python experiment_hamilton_jacobi.py
uv run python experiment_schwarzschild_tetrad.py

With pip on Windows PowerShell:

python -m venv .venv
.\.venv\Scripts\Activate.ps1
pip install -r requirements.txt
python derive_predictions.py
python experiment_phi_gradient.py
python experiment_hamilton_jacobi.py
python experiment_schwarzschild_tetrad.py

With pip on macOS/Linux:

python -m venv .venv
source .venv/bin/activate
pip install -r requirements.txt
python derive_predictions.py
python experiment_phi_gradient.py
python experiment_hamilton_jacobi.py
python experiment_schwarzschild_tetrad.py

License

Code is released under the MIT License. Documentation and explanatory text are released under CC BY 4.0 unless otherwise noted.

See LICENSE and LICENSE-docs.md.