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← Back to Part III: Einstein Field Equations

Chapter 16: Weak Field Limit

The weak field limit connects general relativity to Newtonian gravity and provides the framework for understanding gravitational waves, gravitational lensing, and post-Newtonian corrections to orbital dynamics.

Linearized Gravity

For weak gravitational fields, the metric is a small perturbation of flat spacetime:

\( g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu}, \quad |h_{\mu\nu}| \ll 1 \)

η_μν = Minkowski metric, h_μν = perturbation

To linear order in h, the Einstein tensor becomes:

\( G^{(1)}_{\mu\nu} = \frac{1}{2}\left( \partial^\rho \partial_\mu h_{\nu\rho} + \partial^\rho \partial_\nu h_{\mu\rho} - \Box h_{\mu\nu} - \partial_\mu \partial_\nu h - \eta_{\mu\nu}(\partial^\rho \partial^\sigma h_{\rho\sigma} - \Box h) \right) \)

where h = η^μνh_μν and □ = ∂^μ∂_μ

Gauge Freedom

Infinitesimal coordinate transformations give gauge freedom in h_μν:

\( h_{\mu\nu} \to h_{\mu\nu} + \partial_\mu \xi_\nu + \partial_\nu \xi_\mu \)

Lorenz Gauge (Harmonic)

\( \partial^\mu \bar{h}_{\mu\nu} = 0 \)

where \( \bar{h}_{\mu\nu} = h_{\mu\nu} - \frac{1}{2}\eta_{\mu\nu}h \)

Wave Equation

\( \Box \bar{h}_{\mu\nu} = -\frac{16\pi G}{c^4} T_{\mu\nu} \)

Gravitational waves!

Gravitational Waves

In vacuum (T_μν = 0), the wave equation \( \Box \bar{h}_{\mu\nu} = 0 \) has plane wave solutions:

\( h_{\mu\nu} = \epsilon_{\mu\nu} e^{ik_\rho x^\rho} \)

k^μk_μ = 0 (null wavevector)

Transverse-Traceless (TT) Gauge

For wave propagating in z-direction:

\( h^{TT}_{\mu\nu} = \begin{pmatrix} 0 & 0 & 0 & 0 \\ 0 & h_+ & h_\times & 0 \\ 0 & h_\times & -h_+ & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix} \cos(\omega t - kz) \)

Two polarizations: plus (+) and cross (×)

Interactive Simulation: Gravitational Wave Effects

Run this Python code to visualize how gravitational waves deform a ring of test masses. The simulation shows both plus (+) and cross (×) polarizations at different phases, with the amplitude exaggerated for visibility. Try modifying the parameters!

Gravitational Wave Visualization

Python

Visualize GW effects on test masses in the weak field limit

weak_field_gw.py163 lines

#!/usr/bin/env python3
"""
weak_field_gw.py - Gravitational wave effects in the weak field limit
Visualizes how GWs deform a ring of test masses
"""
import numpy as np
import matplotlib.pyplot as plt

def gw_metric_perturbation(t, h_plus, h_cross, omega):
    """
    Compute TT gauge metric perturbation for GW propagating in z-direction
    """
    phase = omega * t
    h_xx = h_plus * np.cos(phase)
    h_yy = -h_plus * np.cos(phase)
    h_xy = h_cross * np.cos(phase)
    return h_xx, h_yy, h_xy

def deform_circle(theta, h_xx, h_yy, h_xy):
    """
    Compute deformation of unit circle under GW perturbation
    x' = x(1 + h_xx/2) + y*h_xy/2
    y' = y(1 + h_yy/2) + x*h_xy/2
    """
    x = np.cos(theta)
    y = np.sin(theta)
    x_new = x * (1 + h_xx/2) + y * h_xy/2
    y_new = y * (1 + h_yy/2) + x * h_xy/2
    return x_new, y_new

# Physical constants
c = 3e8        # Speed of light (m/s)
G = 6.674e-11  # Gravitational constant
M_sun = 1.989e30  # Solar mass (kg)
pc = 3.086e16  # Parsec (m)

# GW150914-like parameters
M_chirp = 30 * M_sun   # Chirp mass
D = 400e6 * pc         # Distance: 400 Mpc
f_gw = 150             # Peak frequency (Hz)

# Calculate peak strain
h_peak = (4/D) * (G*M_chirp/c**2)**(5/3) * (np.pi*f_gw/c)**(2/3)

print("=" * 70)
print("Weak Field Limit: Gravitational Wave Analysis")
print("=" * 70)
print()
print("━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━")
print("  LINEARIZED GRAVITY")
print("━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━")
print()
print("  Metric perturbation:")
print()
print("  \\( g_{\\mu\\nu} = \\eta_{\\mu\\nu} + h_{\\mu\\nu}, \\quad |h_{\\mu\\nu}| \\ll 1 \\)")
print()
print("  TT gauge metric for wave in z-direction:")
print()
print("  \\( h^{TT}_{\\mu\\nu} = \\begin{pmatrix} 0 & 0 & 0 & 0 \\\\ 0 & h_+ & h_\\times & 0 \\\\ 0 & h_\\times & -h_+ & 0 \\\\ 0 & 0 & 0 & 0 \\end{pmatrix} \\cos(\\omega t - kz) \\)")
print()
print("━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━")
print("  GW150914-LIKE EVENT")
print("━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━")
print()
print(f"  Chirp mass:        \\( \\mathcal{{M}}_c = {M_chirp/M_sun:.0f} \\, M_\\odot \\)")
print(f"  Distance:          \\( D_L = 400 \\) Mpc")
print(f"  Peak frequency:    \\( f = {f_gw} \\) Hz")
print(f"  Peak strain:       \\( h \\approx {h_peak:.2e} \\)")
print()
print("  Strain formula:")
print()
print("  \\( h \\sim \\frac{4}{D_L} \\left(\\frac{G\\mathcal{M}_c}{c^2}\\right)^{5/3} \\left(\\frac{\\pi f}{c}\\right)^{2/3} \\)")
print()

# LIGO sensitivity
L_arm = 4000  # LIGO arm length in meters
delta_L = h_peak * L_arm

print("━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━")
print("  LIGO DETECTION")
print("━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━")
print()
print(f"  Arm length:        \\( L = 4 \\) km")
print(f"  Length change:     \\( \\Delta L = h \\times L \\approx {delta_L:.2e} \\) m")
print()
print("  This is about 1/10,000 of a proton diameter!")
print()
print("  Sensitivity:       \\( h_{\\text{min}} \\sim 10^{-23} \\) (design)")
print()

# Create visualization
omega = 2 * np.pi  # Normalized angular frequency
h_plus_viz = 0.3   # Exaggerated for visibility
theta = np.linspace(0, 2*np.pi, 100)

fig, axes = plt.subplots(2, 3, figsize=(14, 9))

# Plus polarization at different phases
for i, phase in enumerate([0, np.pi/2, np.pi]):
    ax = axes[0, i]
    t = phase / omega
    h_xx, h_yy, h_xy = gw_metric_perturbation(t, h_plus_viz, 0, omega)
    x_new, y_new = deform_circle(theta, h_xx, h_yy, h_xy)

ax.fill(np.cos(theta), np.sin(theta), alpha=0.1, color='gray')
    ax.plot(np.cos(theta), np.sin(theta), 'k--', alpha=0.4, linewidth=1, label='Unperturbed')
    ax.fill(x_new, y_new, alpha=0.3, color='blue')
    ax.plot(x_new, y_new, 'b-', linewidth=2.5, label='Deformed')

# Add test masses
    for angle in np.linspace(0, 2*np.pi, 8, endpoint=False):
        x_m = np.cos(angle) * (1 + h_xx/2 * np.cos(angle)**2 + h_yy/2 * np.sin(angle)**2)
        y_m = np.sin(angle) * (1 + h_yy/2 * np.sin(angle)**2 + h_xx/2 * np.cos(angle)**2)
        ax.plot(x_m, y_m, 'bo', markersize=8)

ax.set_xlim(-1.6, 1.6)
    ax.set_ylim(-1.6, 1.6)
    ax.set_aspect('equal')
    ax.set_title(f'Plus (+) polarization: φ = {phase*180/np.pi:.0f}°', fontsize=12, fontweight='bold')
    ax.grid(True, alpha=0.3)
    ax.axhline(y=0, color='gray', linestyle='-', alpha=0.2)
    ax.axvline(x=0, color='gray', linestyle='-', alpha=0.2)
    if i == 0:
        ax.legend(loc='upper right', fontsize=9)

# Cross polarization at different phases
for i, phase in enumerate([0, np.pi/2, np.pi]):
    ax = axes[1, i]
    t = phase / omega
    h_xx, h_yy, h_xy = gw_metric_perturbation(t, 0, h_plus_viz, omega)
    x_new, y_new = deform_circle(theta, h_xx, h_yy, h_xy)

ax.fill(np.cos(theta), np.sin(theta), alpha=0.1, color='gray')
    ax.plot(np.cos(theta), np.sin(theta), 'k--', alpha=0.4, linewidth=1, label='Unperturbed')
    ax.fill(x_new, y_new, alpha=0.3, color='red')
    ax.plot(x_new, y_new, 'r-', linewidth=2.5, label='Deformed')

ax.set_xlim(-1.6, 1.6)
    ax.set_ylim(-1.6, 1.6)
    ax.set_aspect('equal')
    ax.set_title(f'Cross (×) polarization: φ = {phase*180/np.pi:.0f}°', fontsize=12, fontweight='bold')
    ax.grid(True, alpha=0.3)
    ax.axhline(y=0, color='gray', linestyle='-', alpha=0.2)
    ax.axvline(x=0, color='gray', linestyle='-', alpha=0.2)
    if i == 0:
        ax.legend(loc='upper right', fontsize=9)

plt.suptitle('Gravitational Wave Effect on Ring of Test Masses\n(Amplitude exaggerated for visibility)',
             fontsize=14, fontweight='bold')
plt.tight_layout()
plt.savefig('weak_field_gw.png', dpi=150, bbox_inches='tight')

print("━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━")
print("  TWO POLARIZATION MODES")
print("━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━")
print()
print("  Plus (+):  Stretches along x, compresses along y (and vice versa)")
print("  Cross (×): Stretches along 45°, compresses along 135° (and vice versa)")
print()
print("  General wave is superposition: \\( h = h_+ e_+^{\\mu\\nu} + h_\\times e_\\times^{\\mu\\nu} \\)")
print()
print("✓ Plot saved as 'weak_field_gw.png'")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Fortran: Linearized Gravity Analysis

This Fortran code demonstrates the weak field equations, calculates GW150914-like event parameters, and shows how LIGO detects gravitational waves through tiny length changes.

Weak Field Limit Calculations

Fortran

Linearized gravity and gravitational wave physics

weak_field_limit.f90134 lines

! weak_field_limit.f90 - Weak field approximation calculations
! Demonstrates linearized gravity and gravitational wave physics

program weak_field_limit
    implicit none
    integer, parameter :: dp = selected_real_kind(15)
    real(dp), parameter :: pi = 4.0_dp * atan(1.0_dp)
    real(dp), parameter :: c = 3.0e8_dp      ! m/s
    real(dp), parameter :: G = 6.674e-11_dp  ! N m²/kg²
    real(dp), parameter :: M_sun = 1.989e30_dp  ! kg
    real(dp), parameter :: pc = 3.086e16_dp  ! parsec in meters

real(dp) :: M_chirp, D_L, f_gw, h_strain
    real(dp) :: omega, wavelength, period, E_flux
    real(dp) :: L_arm, delta_L
    real(dp) :: t, dt, t_max, h_plus, h_cross
    integer :: i, n_steps

print '(A)', '╔══════════════════════════════════════════════════════════════════╗'
    print '(A)', '║         Weak Field Limit: Gravitational Wave Analysis            ║'
    print '(A)', '╚══════════════════════════════════════════════════════════════════╝'
    print '(A)', ''

print '(A)', '━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━'
    print '(A)', '  LINEARIZED EINSTEIN EQUATIONS'
    print '(A)', '━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━'
    print '(A)', ''
    print '(A)', '  In the weak field limit, the metric is:'
    print '(A)', ''
    print '(A)', '  \( g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu}, \quad |h_{\mu\nu}| \ll 1 \)'
    print '(A)', ''
    print '(A)', '  The linearized Einstein tensor:'
    print '(A)', ''
    print '(A)', '  \( G^{(1)}_{\mu\nu} = \frac{1}{2}\left( \partial^\rho \partial_\mu h_{\nu\rho} + \partial^\rho \partial_\nu h_{\mu\rho} - \Box h_{\mu\nu} - \partial_\mu \partial_\nu h \right) \)'
    print '(A)', ''
    print '(A)', '  In Lorenz gauge \( \partial^\mu \bar{h}_{\mu\nu} = 0 \):'
    print '(A)', ''
    print '(A)', '  \( \Box \bar{h}_{\mu\nu} = -\frac{16\pi G}{c^4} T_{\mu\nu} \)'
    print '(A)', ''

! GW150914 parameters
    M_chirp = 30.0_dp * M_sun
    D_L = 400.0_dp * 1.0e6_dp * pc
    f_gw = 150.0_dp

! Calculate strain
    h_strain = (4.0_dp/D_L) * (G*M_chirp/c**2)**(5.0_dp/3.0_dp) * &
               (pi*f_gw/c)**(2.0_dp/3.0_dp)

omega = 2.0_dp * pi * f_gw
    wavelength = c / f_gw
    period = 1.0_dp / f_gw

print '(A)', '━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━'
    print '(A)', '  GW150914-LIKE EVENT PARAMETERS'
    print '(A)', '━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━'
    print '(A)', ''
    print '(A,F8.1,A)', '  Chirp mass:    \( \mathcal{M}_c = ', M_chirp/M_sun, ' \, M_\odot \)'
    print '(A,F8.1,A)', '  Distance:      \( D_L = ', D_L/(1.0e6_dp*pc), ' \) Mpc'
    print '(A,F8.1,A)', '  Frequency:     \( f = ', f_gw, ' \) Hz'
    print '(A,E12.4,A)', '  Wavelength:    \( \lambda = ', wavelength, ' \) m'
    print '(A,E12.4,A)', '  Period:        \( T = ', period, ' \) s'
    print '(A)', ''
    print '(A,E12.4)', '  Peak strain:   \( h \approx \)', h_strain
    print '(A)', ''

! LIGO detection
    L_arm = 4000.0_dp
    delta_L = h_strain * L_arm

print '(A)', '━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━'
    print '(A)', '  LIGO INTERFEROMETER'
    print '(A)', '━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━'
    print '(A)', ''
    print '(A)', '  Detection principle:'
    print '(A)', ''
    print '(A)', '  \( \Delta L = h \times L \)'
    print '(A)', ''
    print '(A,F8.1,A)', '  Arm length:        \( L = ', L_arm/1000.0_dp, ' \) km'
    print '(A,E12.4,A)', '  Length change:     \( \Delta L \approx ', delta_L, ' \) m'
    print '(A)', ''
    print '(A)', '  For comparison:'
    print '(A,E12.4,A)', '    Proton diameter: \( d_p \approx ', 1.7e-15_dp, ' \) m'
    print '(A,F8.4,A)', '    Ratio:           \( \Delta L / d_p \approx ', delta_L/1.7e-15_dp, ' \)'
    print '(A)', ''

! Energy flux
    h_plus = h_strain
    h_cross = 0.5_dp * h_strain
    E_flux = (c**3/(32.0_dp*pi*G)) * omega**2 * (h_plus**2 + h_cross**2)

print '(A)', '━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━'
    print '(A)', '  GRAVITATIONAL WAVE ENERGY'
    print '(A)', '━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━'
    print '(A)', ''
    print '(A)', '  Energy flux formula:'
    print '(A)', ''
    print '(A)', '  \( F = \frac{c^3}{32\pi G} \omega^2 \left( h_+^2 + h_\times^2 \right) \)'
    print '(A)', ''
    print '(A,E12.4,A)', '  Energy flux:   \( F \approx ', E_flux, ' \) W/m\( ^2 \)'
    print '(A)', ''

! Wave propagation table
    print '(A)', '━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━'
    print '(A)', '  WAVE PROPAGATION (One Period)'
    print '(A)', '━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━'
    print '(A)', ''
    print '(A)', '  \( t \) (ms)      \( h_+ \)           \( h_\times \)'
    print '(A)', '  ─────────────────────────────────────────────'

t_max = period
    n_steps = 10
    dt = t_max / n_steps

do i = 0, n_steps
        t = i * dt
        print '(F10.4, 2E14.4)', t*1000, &
            h_plus * cos(omega * t), &
            h_cross * sin(omega * t)
    end do

print '(A)', ''
    print '(A)', '┌────────────────────────────────────────────────────────────────────┐'
    print '(A)', '│  KEY PHYSICS                                                       │'
    print '(A)', '├────────────────────────────────────────────────────────────────────┤'
    print '(A)', '│  • GWs are transverse: oscillations perpendicular to propagation   │'
    print '(A)', '│  • Two polarizations: + and × (rotated by 45°)                     │'
    print '(A)', '│  • Travel at speed of light: \( v = c \)                            │'
    print '(A)', '│  • Quadrupole radiation: \( h \propto \ddot{Q}_{ij} / r \)           │'
    print '(A)', '└────────────────────────────────────────────────────────────────────┘'
    print '(A)', ''

end program weak_field_limit

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Code will be compiled with gfortran and executed on the server

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