General Relativity

The general relativity domain provides tools for calculations in curved spacetime, including black holes, gravitational lensing, and cosmology.

Overview 

The general relativity module implements:

Schwarzschild metric for non-rotating black holes
Kerr metric for rotating black holes
Geodesic solver for particle/photon trajectories
Gravitational lensing calculations
FLRW cosmology for the expanding universe

Quick Start 

from mechanics_dsl.domains.general_relativity import (
    SchwarzschildMetric, GravitationalLensing, FLRWCosmology,
    SOLAR_MASS
)

# Black hole analysis
bh = SchwarzschildMetric(mass=10 * SOLAR_MASS)
rs = bh.schwarzschild_radius()  # ~30 km
r_isco = bh.isco_radius()       # 3 × rs
T_hawking = bh.hawking_temperature()

# Gravitational lensing
lens = GravitationalLensing(mass=SOLAR_MASS)
alpha = lens.deflection_angle(impact_parameter=7e8)  # 1.75 arcsec

# Cosmology
cosmos = FLRWCosmology(H0=70, Omega_m=0.3, Omega_Lambda=0.7)
age = cosmos.age()

Classes 

SchwarzschildMetric 

The Schwarzschild metric describes non-rotating, spherically symmetric black holes:

\[ds^2 = -\left(1 - \frac{r_s}{r}\right)c^2 dt^2 + \left(1 - \frac{r_s}{r}\right)^{-1} dr^2 + r^2 d\Omega^2\]

where \(r_s = 2GM/c^2\) is the Schwarzschild radius.

Key methods:

schwarzschild_radius(): Event horizon radius
isco_radius(): Innermost stable circular orbit (3rs)
photon_sphere_radius(): Light orbit radius (1.5rs)
hawking_temperature(): Black hole temperature
gravitational_redshift(r): Photon redshift from radius r

KerrMetric 

The Kerr metric describes rotating black holes with spin parameter \(a = J/(Mc)\).

Key features:

outer_horizon(): Outer event horizon
inner_horizon(): Inner (Cauchy) horizon
ergosphere_radius(theta): Static limit surface
frame_dragging_rate(r): Spacetime rotation rate
isco_radius(prograde): ISCO for prograde/retrograde orbits

GravitationalLensing 

Light deflection by massive objects.

Deflection angle (weak field):

\[\alpha = \frac{4GM}{c^2 b} = \frac{2r_s}{b}\]

For the Sun at grazing incidence: α ≈ 1.75 arcseconds.

FLRWCosmology 

Friedmann-Lemaître-Robertson-Walker cosmological model:

\[ds^2 = -c^2 dt^2 + a(t)^2 \left[\frac{dr^2}{1-kr^2} + r^2 d\Omega^2\right]\]

Key methods:

hubble_parameter(z): H(z) at redshift z
age(): Age of the universe
comoving_distance(z): Distance to redshift z
luminosity_distance(z): For standard candles

Physical Constants 

from mechanics_dsl.domains.general_relativity import (
    SPEED_OF_LIGHT,      # 299792458 m/s
    GRAVITATIONAL_CONSTANT,  # 6.674e-11 m³/(kg·s²)
    SOLAR_MASS           # 1.989e30 kg
)