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← Back to Part IV: Classic Solutions

Chapter 22: Gravitational Waves

Gravitational waves are ripples in spacetime that propagate at the speed of light. Predicted by Einstein in 1916 and first directly detected by LIGO in 2015, they've opened a new window on the universe.

Wave Equation

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

In vacuum: □h̄_μν = 0 (wave equation)

Gravitational waves travel at c and carry energy away from accelerating masses. For a binary system, the energy loss causes orbital decay.

Quadrupole Radiation

\( h_{ij}^{TT} = \frac{2G}{c^4 r} \ddot{Q}_{ij}^{TT} \)

Q_ij = quadrupole moment tensor

Power Radiated

\( P = \frac{G}{5c^5} \langle \dddot{Q}_{ij} \dddot{Q}^{ij} \rangle \)

For circular binary: P ∝ (Mω)^(10/3)

Interactive Simulation: Binary Inspiral Waveform

Run this Python code to simulate a gravitational wave signal from a binary black hole merger similar to GW150914, the first directly detected gravitational wave event. Try modifying the masses to see how the waveform changes!

GW150914-like Inspiral Waveform

Python

Simulate gravitational waves from binary black hole merger

gw_waveform.py98 lines

#!/usr/bin/env python3
"""
gw_waveform.py - Gravitational wave waveform from binary inspiral
Simulates GW150914-like event
"""
import numpy as np
import matplotlib.pyplot as plt

# 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)

def chirp_mass(m1, m2):
    """Chirp mass: determines GW amplitude and frequency evolution"""
    return (m1 * m2)**(3/5) / (m1 + m2)**(1/5)

def gw_strain(t, M_c, D, t_c):
    """Leading-order inspiral waveform (Newtonian approximation)"""
    tau = t_c - t
    tau = np.maximum(tau, 1e-6)  # Avoid division by zero

# Frequency evolution (chirp)
    f = (5**(3/8) / (8*np.pi)) * (G*M_c/c**3)**(-5/8) * tau**(-3/8)

# Phase evolution
    phi = -2 * (5*G*M_c/(c**3))**(-5/8) * tau**(5/8)

# Strain amplitude
    h = (4/D) * (G*M_c/c**2)**(5/4) * (np.pi*f/c)**(2/3) * np.cos(phi)

return h, f

# GW150914-like event parameters
m1 = 36 * M_sun   # First black hole mass
m2 = 29 * M_sun   # Second black hole mass
D = 410e6 * pc    # Distance: 410 Mpc
M_c = chirp_mass(m1, m2)

t_c = 0.4  # Coalescence time
t = np.linspace(0, t_c - 0.001, 10000)

# Generate waveform
h, f = gw_strain(t, M_c, D, t_c)

# Create figure with two panels
fig, axes = plt.subplots(2, 1, figsize=(12, 8))

# Top panel: Strain waveform
ax1 = axes[0]
ax1.plot(t, h * 1e21, 'b-', linewidth=0.5)
ax1.set_xlabel('Time before coalescence (s)', fontsize=12)
ax1.set_ylabel('Strain h (×10⁻²¹)', fontsize=12)
ax1.set_title('GW150914-like Inspiral Waveform', fontsize=14)
ax1.grid(True, alpha=0.3)
ax1.axhline(y=0, color='gray', linestyle='-', alpha=0.3)

# Add annotation for chirp
ax1.annotate('Chirp: amplitude & frequency\nincrease as merger approaches',
             xy=(0.35, max(h)*1e21*0.7), fontsize=10, color='blue')

# Bottom panel: Frequency evolution
ax2 = axes[1]
ax2.plot(t, f, 'r-', linewidth=1.5)
ax2.set_xlabel('Time before coalescence (s)', fontsize=12)
ax2.set_ylabel('Frequency (Hz)', fontsize=12)
ax2.set_title('Chirp: Frequency Evolution', fontsize=14)
ax2.grid(True, alpha=0.3)
ax2.set_ylim(0, 500)
ax2.fill_between(t, 0, f, alpha=0.2, color='red')

# Mark LIGO sensitive band
ax2.axhline(y=20, color='green', linestyle='--', alpha=0.7, label='LIGO lower cutoff (20 Hz)')
ax2.axhline(y=300, color='orange', linestyle='--', alpha=0.7, label='Peak sensitivity')
ax2.legend(loc='upper left', fontsize=9)

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

# Print results
print("=" * 55)
print("Gravitational Wave Inspiral Simulation")
print("=" * 55)
print(f"\nBinary Parameters (GW150914-like):")
print(f"  Black hole 1: {m1/M_sun:.0f} solar masses")
print(f"  Black hole 2: {m2/M_sun:.0f} solar masses")
print(f"  Chirp mass: {M_c/M_sun:.1f} solar masses")
print(f"  Distance: 410 Mpc")
print(f"\nWaveform Properties:")
print(f"  Initial frequency: {f[0]:.1f} Hz")
print(f"  Peak frequency: {f[-1]:.0f} Hz")
print(f"  Peak strain: {max(abs(h)):.2e}")
print(f"\nDetection:")
print(f"  LIGO sensitivity: ~10⁻²³ strain")
print(f"  Signal well above threshold!")
print("\n✓ Plot saved as 'gw_waveform.png'")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

← FLRW Cosmology Next: PP Waves →