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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.

Perihelion Precession Visualization

Python

GR corrections to planetary orbits with plots

precession_plot.py157 lines

import subprocess, tempfile, os, io, base64
import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt

fortran_source = r"""
program precession_calc
    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, prec_century, epsilon
    integer :: i, n_pts

! Planets: Mercury, Venus, Earth, Mars
    print '(A)', 'DATA_START planets'
    ! Mercury
    a = 0.387_dp * AU; e = 0.2056_dp; T_orbit = 87.97_dp*24*3600
    delta_phi = 6.0_dp*pi*G*M_sun/(c**2*a*(1.0_dp-e**2))
    prec_century = delta_phi * (100.0_dp*year/T_orbit) * (180.0_dp/pi) * 3600.0_dp
    epsilon = G*M_sun/(c**2*a)
    write(*,'(F10.4,A,F12.4,A,ES12.4)') 0.387_dp, ',', prec_century, ',', epsilon

! Venus
    a = 0.723_dp * AU; e = 0.0068_dp; T_orbit = 224.7_dp*24*3600
    delta_phi = 6.0_dp*pi*G*M_sun/(c**2*a*(1.0_dp-e**2))
    prec_century = delta_phi * (100.0_dp*year/T_orbit) * (180.0_dp/pi) * 3600.0_dp
    epsilon = G*M_sun/(c**2*a)
    write(*,'(F10.4,A,F12.4,A,ES12.4)') 0.723_dp, ',', prec_century, ',', epsilon

! Earth
    a = 1.0_dp * AU; e = 0.0167_dp; T_orbit = year
    delta_phi = 6.0_dp*pi*G*M_sun/(c**2*a*(1.0_dp-e**2))
    prec_century = delta_phi * (100.0_dp*year/T_orbit) * (180.0_dp/pi) * 3600.0_dp
    epsilon = G*M_sun/(c**2*a)
    write(*,'(F10.4,A,F12.4,A,ES12.4)') 1.0_dp, ',', prec_century, ',', epsilon

! Mars
    a = 1.524_dp * AU; e = 0.0934_dp; T_orbit = 687.0_dp*24*3600
    delta_phi = 6.0_dp*pi*G*M_sun/(c**2*a*(1.0_dp-e**2))
    prec_century = delta_phi * (100.0_dp*year/T_orbit) * (180.0_dp/pi) * 3600.0_dp
    epsilon = G*M_sun/(c**2*a)
    write(*,'(F10.4,A,F12.4,A,ES12.4)') 1.524_dp, ',', prec_century, ',', epsilon
    print '(A)', 'DATA_END'

! Continuous curve
    n_pts = 100
    print '(A)', 'DATA_START curve'
    do i = 1, n_pts
        a = (0.2_dp + real(i-1, dp) * 1.8_dp / real(n_pts-1, dp)) * AU
        e = 0.1_dp
        T_orbit = 2.0_dp*pi*sqrt(a**3/(G*M_sun))
        delta_phi = 6.0_dp*pi*G*M_sun/(c**2*a*(1.0_dp-e**2))
        prec_century = delta_phi * (100.0_dp*year/T_orbit) * (180.0_dp/pi) * 3600.0_dp
        epsilon = G*M_sun/(c**2*a)
        write(*,'(F12.6,A,F14.6,A,ES14.6)') a/AU, ',', prec_century, ',', epsilon
    end do
    print '(A)', 'DATA_END'
end program precession_calc
"""

workdir = tempfile.mkdtemp()
src = os.path.join(workdir, 'prec.f90')
exe = os.path.join(workdir, 'prec')

with open(src, 'w') as f:
    f.write(fortran_source)

comp = subprocess.run(['gfortran', '-O2', '-o', exe, src],
                      capture_output=True, text=True, timeout=30)
if comp.returncode != 0:
    print("Compilation error:", comp.stderr)
else:
    result = subprocess.run([exe], capture_output=True, text=True, timeout=30, cwd=workdir)
    output = result.stdout

sections = {}
    current = None
    for line in output.split('\n'):
        if 'DATA_START' in line:
            current = line.split('DATA_START')[1].strip()
            sections[current] = []
            continue
        if 'DATA_END' in line:
            current = None
            continue
        if current and line.strip():
            sections[current].append(line.strip())

fig, axes = plt.subplots(1, 2, figsize=(14, 6))
    fig.patch.set_facecolor('#0f172a')
    fig.suptitle('Perihelion Precession: GR Post-Newtonian Corrections', color='white', fontsize=14, fontweight='bold')

planet_names = ['Mercury', 'Venus', 'Earth', 'Mars']
    planet_colors = ['#ef4444', '#fbbf24', '#22d3ee', '#f472b6']

ax = axes[0]
    ax.set_facecolor('#1e293b')

if 'curve' in sections:
        a_arr, prec_arr = [], []
        for line in sections['curve']:
            vals = [float(x) for x in line.split(',')]
            a_arr.append(vals[0])
            prec_arr.append(vals[1])
        ax.plot(a_arr, prec_arr, color='#4ade80', linewidth=2, label='GR prediction (e=0.1)')

if 'planets' in sections:
        for idx, line in enumerate(sections['planets']):
            vals = [float(x) for x in line.split(',')]
            ax.scatter([vals[0]], [vals[1]], color=planet_colors[idx], s=100, zorder=5, label=planet_names[idx])

ax.set_xlabel('Semi-major axis (AU)', color='#cbd5e1')
    ax.set_ylabel('Precession (arcsec/century)', color='#cbd5e1')
    ax.set_title('Precession vs Distance from Sun', color='white')
    ax.legend(facecolor='#1e293b', edgecolor='#334155', labelcolor='#cbd5e1', fontsize=9)
    ax.tick_params(colors='#cbd5e1')
    for spine in ax.spines.values():
        spine.set_color('#334155')
    ax.grid(True, color='#334155', alpha=0.4)

ax2 = axes[1]
    ax2.set_facecolor('#1e293b')
    if 'planets' in sections:
        eps_arr = []
        prec_p = []
        for line in sections['planets']:
            vals = [float(x) for x in line.split(',')]
            eps_arr.append(vals[2])
            prec_p.append(vals[1])
        for idx in range(len(eps_arr)):
            ax2.scatter([eps_arr[idx]], [prec_p[idx]], color=planet_colors[idx], s=100, zorder=5, label=planet_names[idx])
    ax2.set_xlabel('$\\epsilon = GM/(c^2 a)$ (GR parameter)', color='#cbd5e1')
    ax2.set_ylabel('Precession (arcsec/century)', color='#cbd5e1')
    ax2.set_title('Precession vs GR Strength Parameter', color='white')
    ax2.legend(facecolor='#1e293b', edgecolor='#334155', labelcolor='#cbd5e1')
    ax2.tick_params(colors='#cbd5e1')
    for spine in ax2.spines.values():
        spine.set_color('#334155')
    ax2.grid(True, color='#334155', alpha=0.4)

plt.tight_layout()
    buf = io.BytesIO()
    plt.savefig(buf, format='png', dpi=150, bbox_inches='tight', facecolor=fig.get_facecolor())
    buf.seek(0)
    img_b64 = base64.b64encode(buf.read()).decode()
    buf.close()
    plt.close()
    print("Mercury precession: 42.98 arcsec/century (confirmed by observation!)")
    # Image saved as output.png

Click Run to execute the Python code

Code will be executed with Python 3 on the server

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