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← Back to Part III: Einstein Field Equations

Chapter 17: Newtonian Limit

General relativity must reduce to Newtonian gravity in the appropriate limit: weak fields, slow motions, and non-relativistic matter. This correspondence shows that Einstein's theory is the correct generalization of Newton's.

Newtonian Limit Conditions

Weak Field

$ |h_{\mu\nu}| \ll 1 $

|Φ|/c² ≪ 1 where Φ is Newtonian potential

Slow Motion

$ v \ll c $

Non-relativistic velocities

Static or Slow-Varying Field

$ \partial_t g_{\mu\nu} \approx 0 $

Time derivatives negligible

Non-Relativistic Matter

$ T_{00} \approx \rho c^2 \gg T_{ij} $

Pressure negligible vs rest mass energy

Derivation

With g₀₀ = -(1 + 2Φ/c²), the 00-component of Einstein's equations:

$ G_{00} = \frac{8\pi G}{c^4} T_{00} $

↓

$ \nabla^2 h_{00} = -\frac{16\pi G}{c^4} \rho c^2 $

↓

$ \nabla^2 \Phi = 4\pi G \rho $

Poisson's equation! (Newtonian gravity)

The geodesic equation for slow motion (u^μ ≈ (c, 0, 0, 0)) gives:

$ \frac{d^2 \vec{x}}{dt^2} = -\nabla \Phi $

Newton's law of gravitation!

Post-Newtonian Corrections

Beyond the Newtonian limit, GR predicts corrections of order (v/c)² and (Φ/c²):

Perihelion Precession

$ \Delta\phi = \frac{6\pi GM}{c^2 a(1-e^2)} $ per orbit

Mercury: 43 arcsec/century (confirmed!)

Light Deflection

$ \delta\theta = \frac{4GM}{c^2 b} $ (twice Newtonian prediction)

Sun: 1.75 arcsec (confirmed 1919!)

Shapiro Time Delay

$ \Delta t = \frac{4GM}{c^3} \ln\left(\frac{4r_1 r_2}{b^2}\right) $

Radar ranging to planets, pulsars

Interactive Simulation: Newtonian Limit

Run this Python code to explore the transition from GR to Newtonian gravity. The simulation computes the relativistic parameter ε = |Φ|/c² for different objects and visualizes post-Newtonian corrections like Mercury's perihelion precession.

Newtonian Limit Analysis

Python

Explore when GR reduces to Newtonian gravity

newtonian_limit.py165 lines

#!/usr/bin/env python3
"""
newtonian_limit.py - Verify Newtonian limit of general relativity
Visualizes the transition from GR to Newtonian gravity
"""
import numpy as np
import matplotlib.pyplot as plt

# Physical constants
G = 6.674e-11  # N m²/kg²
c = 3e8  # m/s
M_sun = 1.989e30  # kg
M_earth = 5.972e24  # kg
R_earth = 6.371e6  # m
AU = 1.496e11  # m

def newtonian_potential(M, r):
    """Newtonian gravitational potential"""
    return -G * M / r

def metric_component_g00(M, r):
    """g_00 component in weak field"""
    Phi = newtonian_potential(M, r)
    return -(1 + 2*Phi/c**2)

def relativistic_parameter(M, r):
    """Measure of relativistic effects: |Φ|/c²"""
    return abs(newtonian_potential(M, r)) / c**2

print("=" * 70)
print("Newtonian Limit of General Relativity")
print("=" * 70)
print()
print("━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━")
print("  FUNDAMENTAL CORRESPONDENCE")
print("━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━")
print()
print("  Weak field metric:")
print()
print("  \$ g_{00} = -\\left(1 + \\frac{2\\Phi}{c^2}\\right), \\quad |\\Phi|/c^2 \\ll 1 \$")
print()
print("  Einstein equations reduce to Poisson's equation:")
print()
print("  \$ \\nabla^2 \\Phi = 4\\pi G \\rho \$")
print()
print("  Geodesic equation gives Newton's 2nd law:")
print()
print("  \$ \\frac{d^2 \\vec{x}}{dt^2} = -\\nabla \\Phi \$")
print()

# Analyze different regimes
objects = [
    ("Earth surface", M_earth, R_earth),
    ("Sun surface", M_sun, 6.96e8),
    ("White dwarf", 1.0*M_sun, 0.01*6.96e8),
    ("Neutron star", 1.4*M_sun, 10e3),
    ("Near BH (3Rs)", M_sun, 3*2*G*M_sun/c**2)
]

print("━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━")
print("  RELATIVISTIC PARAMETER \$ \\epsilon = |\\Phi|/c^2 \$")
print("━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━")
print()

epsilons = []
names = []
for name, M, r in objects:
    eps = relativistic_parameter(M, r)
    g00 = metric_component_g00(M, r)
    epsilons.append(eps)
    names.append(name)

print(f"  {name}:")
    print(f"    \$ \\epsilon = |\\Phi|/c^2 = {eps:.2e} \$")
    print(f"    \$ g_{{00}} = {g00:.8f} \$")
    if eps < 1e-6:
        print(f"    → Newtonian limit excellent")
    elif eps < 1e-3:
        print(f"    → Small GR corrections")
    elif eps < 0.1:
        print(f"    → Significant GR effects")
    else:
        print(f"    → Strong field: full GR required!")
    print()

# Post-Newtonian effects
print("━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━")
print("  POST-NEWTONIAN CORRECTIONS")
print("━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━")
print()

# Mercury perihelion precession
a_merc = 5.79e10  # semi-major axis (m)
e_merc = 0.2056   # eccentricity
T_merc = 88 * 24 * 3600  # orbital period (s)

delta_phi = 6 * np.pi * G * M_sun / (c**2 * a_merc * (1 - e_merc**2))
orbits_per_century = 100 * 365.25 * 24 * 3600 / T_merc
precession_arcsec = delta_phi * orbits_per_century * (180/np.pi) * 3600

print("  Mercury Perihelion Precession:")
print()
print("  \$ \\Delta\\phi = \\frac{6\\pi GM}{c^2 a(1-e^2)} \$ per orbit")
print()
print(f"    Per orbit: \$ \\Delta\\phi = {delta_phi:.6e} \$ rad")
print(f"    Per century: \$ {precession_arcsec:.2f} \$ arcsec")
print(f"    Observed: 43 arcsec/century ✓")
print()

# Light deflection
r_sun = 6.96e8
theta_deflect = 4 * G * M_sun / (c**2 * r_sun)
theta_arcsec = theta_deflect * (180/np.pi) * 3600

print("  Light Deflection by Sun:")
print()
print("  \$ \\delta\\theta = \\frac{4GM}{c^2 b} \$")
print()
print(f"    GR prediction: {theta_arcsec:.4f} arcsec")
print(f"    Newtonian: {theta_arcsec/2:.4f} arcsec")
print("    → GR is TWICE Newton (confirmed 1919!)")
print()

# Create visualization
fig, axes = plt.subplots(1, 2, figsize=(14, 6))

# Left plot: Relativistic parameter for different objects
ax1 = axes[0]
colors = ['green', 'blue', 'orange', 'red', 'purple']
bars = ax1.barh(range(len(epsilons)), np.log10(epsilons), color=colors)
ax1.set_yticks(range(len(names)))
ax1.set_yticklabels(names)
ax1.set_xlabel(r'$\log_{10}(|\Phi|/c^2)$', fontsize=12)
ax1.set_title('Relativistic Parameter by Object', fontsize=14, fontweight='bold')
ax1.axvline(x=-6, color='green', linestyle='--', alpha=0.7, label='Newtonian limit')
ax1.axvline(x=-3, color='orange', linestyle='--', alpha=0.7, label='GR corrections')
ax1.axvline(x=-1, color='red', linestyle='--', alpha=0.7, label='Strong field')
ax1.legend(loc='lower right', fontsize=9)
ax1.grid(True, alpha=0.3, axis='x')

# Add value annotations
for i, (bar, eps) in enumerate(zip(bars, epsilons)):
    ax1.text(bar.get_width() + 0.1, bar.get_y() + bar.get_height()/2,
             f'{eps:.1e}', va='center', fontsize=9)

# Right plot: g_00 vs radius for Sun
ax2 = axes[1]
r_range = np.linspace(r_sun, 100*r_sun, 500)
g00_vals = [metric_component_g00(M_sun, r) for r in r_range]
deviation = [abs(g + 1) for g in g00_vals]

ax2.semilogy(r_range/r_sun, deviation, 'b-', linewidth=2)
ax2.set_xlabel(r'$r/R_{\odot}$', fontsize=12)
ax2.set_ylabel(r'$|g_{00} + 1| = 2|\Phi|/c^2$', fontsize=12)
ax2.set_title('Metric Deviation from Flat Space (Sun)', fontsize=14, fontweight='bold')
ax2.grid(True, alpha=0.3)
ax2.axhline(y=1e-6, color='green', linestyle='--', alpha=0.7, label='Newtonian regime')
ax2.fill_between(r_range/r_sun, 0, deviation, alpha=0.2, color='blue')
ax2.legend(fontsize=10)

plt.tight_layout()
plt.savefig('newtonian_limit.png', dpi=150, bbox_inches='tight')

print("✓ Plot saved as 'newtonian_limit.png'")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Fortran: Orbital Precession

This Fortran code calculates the GR perihelion precession for all inner planets and the famous binary pulsar PSR B1913+16, demonstrating the post-Newtonian regime.

Post-Newtonian Precession

Fortran

Calculate GR corrections to planetary orbits

newtonian_limit.f90122 lines

! newtonian_limit.f90 - Orbital precession in the post-Newtonian regime
! Demonstrates GR corrections to Newtonian gravity

program newtonian_limit
    implicit none
    integer, parameter :: dp = selected_real_kind(15)
    real(dp), parameter :: pi = 4.0_dp * atan(1.0_dp)
    real(dp), parameter :: G = 6.674e-11_dp
    real(dp), parameter :: c = 3.0e8_dp
    real(dp), parameter :: M_sun = 1.989e30_dp
    real(dp), parameter :: AU = 1.496e11_dp
    real(dp), parameter :: year = 365.25_dp * 24 * 3600

real(dp) :: a, e, T_orbit, delta_phi, precession_century
    real(dp) :: epsilon, r_s

print '(A)', '╔══════════════════════════════════════════════════════════════════╗'
    print '(A)', '║             Newtonian Limit and Post-Newtonian Corrections       ║'
    print '(A)', '╚══════════════════════════════════════════════════════════════════╝'
    print '(A)', ''

print '(A)', '━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━'
    print '(A)', '  NEWTONIAN LIMIT CONDITIONS'
    print '(A)', '━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━'
    print '(A)', ''
    print '(A)', '  GR reduces to Newtonian gravity when:'
    print '(A)', ''
    print '(A)', '  1. Weak field:    $ |h_{\mu\nu}| \ll 1 $'
    print '(A)', '  2. Slow motion:   $ v \ll c $'
    print '(A)', '  3. Static field:  $ \partial_t g_{\mu\nu} \approx 0 $'
    print '(A)', ''
    print '(A)', '  Key correspondence:'
    print '(A)', ''
    print '(A)', '  $ g_{00} = -\left(1 + \frac{2\Phi}{c^2}\right) \quad \Rightarrow \quad \nabla^2\Phi = 4\pi G\rho $'
    print '(A)', ''

! Schwarzschild radius
    r_s = 2.0_dp * G * M_sun / c**2

print '(A)', '━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━'
    print '(A)', '  PERIHELION PRECESSION'
    print '(A)', '━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━'
    print '(A)', ''
    print '(A)', '  GR predicts precession:'
    print '(A)', ''
    print '(A)', '  $ \Delta\phi = \frac{6\pi GM}{c^2 a(1-e^2)} $ per orbit'
    print '(A)', ''
    print '(A)', '  Planet       a (AU)    e        Precession ("/century)'
    print '(A)', '  ─────────────────────────────────────────────────────────'

! Mercury
    a = 0.387_dp * AU
    e = 0.2056_dp
    T_orbit = 87.97_dp * 24 * 3600
    call compute_precession(a, e, T_orbit, delta_phi, precession_century)
    epsilon = G * M_sun / (c**2 * a)
    print '(A10, F8.3, F9.4, F14.2, A)', 'Mercury', a/AU, e, precession_century, '  ← confirmed!'

! Venus
    a = 0.723_dp * AU
    e = 0.0068_dp
    T_orbit = 224.7_dp * 24 * 3600
    call compute_precession(a, e, T_orbit, delta_phi, precession_century)
    print '(A10, F8.3, F9.4, F14.2)', 'Venus', a/AU, e, precession_century

! Earth
    a = 1.0_dp * AU
    e = 0.0167_dp
    T_orbit = year
    call compute_precession(a, e, T_orbit, delta_phi, precession_century)
    print '(A10, F8.3, F9.4, F14.2)', 'Earth', a/AU, e, precession_century

! Mars
    a = 1.524_dp * AU
    e = 0.0934_dp
    T_orbit = 687.0_dp * 24 * 3600
    call compute_precession(a, e, T_orbit, delta_phi, precession_century)
    print '(A10, F8.3, F9.4, F14.2)', 'Mars', a/AU, e, precession_century

print '(A)', ''

! Binary pulsar
    print '(A)', '━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━'
    print '(A)', '  BINARY PULSAR PSR B1913+16'
    print '(A)', '━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━'
    print '(A)', ''
    a = 1.95e9_dp
    e = 0.617_dp
    T_orbit = 7.75_dp * 3600
    call compute_precession(a, e, T_orbit, delta_phi, precession_century)
    print '(A,F10.4,A)', '  Precession: $ \Delta\phi = ', delta_phi * 180/pi, '^\circ $ /orbit'
    print '(A)', ''
    print '(A)', '  ⟹  4.2° per year (observed & confirmed!)'
    print '(A)', ''
    print '(A)', '  This binary also confirms gravitational wave emission'
    print '(A)', '  through orbital decay (Nobel Prize 1993).'
    print '(A)', ''

print '(A)', '┌────────────────────────────────────────────────────────────────────┐'
    print '(A)', '│  HISTORICAL SIGNIFICANCE                                          │'
    print '(A)', '├────────────────────────────────────────────────────────────────────┤'
    print '(A)', '│  Mercury''s anomalous precession (43"/century) was unexplained    │'
    print '(A)', '│  for 60 years until Einstein calculated it from GR in 1915.       │'
    print '(A)', '│  This was the first empirical success of general relativity!      │'
    print '(A)', '└────────────────────────────────────────────────────────────────────┘'
    print '(A)', ''

contains

subroutine compute_precession(a, e, T_orbit, delta_phi, precession_century)
        real(dp), intent(in) :: a, e, T_orbit
        real(dp), intent(out) :: delta_phi, precession_century
        real(dp) :: orbits_century

delta_phi = 6.0_dp * pi * G * M_sun / (c**2 * a * (1.0_dp - e**2))
        orbits_century = 100.0_dp * year / T_orbit
        precession_century = delta_phi * orbits_century * (180.0_dp/pi) * 3600.0_dp

end subroutine

end program newtonian_limit

Click Run to execute the Fortran code

Code will be compiled with gfortran and executed on the server

← Weak Field Limit Next: Part IV - Classic Solutions →