Gravitational Waves

Gravitational waves are ripples in the fabric of spacetime, predicted by GR and first directly detected by LIGO in 2015. They are produced by accelerating masses and travel at the speed of light. Binary neutron star and black hole mergers are the most prominent sources.

Key Concepts

Weak-field: g_μν = η_μν + h_μν with |h_μν| << 1
Transverse-traceless gauge: two polarization modes h₊ and h×
GW strain: h = ΔL/L
Quadrupole formula: P = (G/5c⁵)(d³Q_ij/dt³)²
Chirp mass: determines frequency evolution of binary inspiral

Key Equations

Linearized GR metric

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

GW strain

h = \frac{\Delta L}{L}

Quadrupole power

P = -\frac{G}{5c^5}\left\langle\ddot{Q}_{ij}\ddot{Q}^{ij}\right\rangle

Chirp mass

\mathcal{M} = \frac{(m_1 m_2)^{3/5}}{(m_1+m_2)^{1/5}}

Worked Example

Example Problem

Problem

LIGO detected h~10⁻²¹ over a distance L=4 km. Find ΔL.

Solution

ΔL = h×L = 10⁻²¹×4000 m = 4×10⁻¹⁸ m ≈ 1/1000 of a proton radius.

Practice

Exercises

7 problems

1 of 7

LIGO arm length L=4.0 km, GW strain h=5×10⁻²². Find ΔL in meters.

Your answer

2 of 7

GW150914 involved two BHs each of ~35 M_sun. Find the chirp mass M_c = (m₁m₂)^{3/5}/(m₁+m₂)^{1/5} for m₁=m₂=35 M_sun in M_sun.

Your answer

M_sun

3 of 7

The frequency of GWs from a circular binary is f_GW = 2f_orbital. For two equal masses m=10 M_sun orbiting at separation r=100 km, find f_orbital using f = √(GM_tot/r³)/(2π) in Hz.

Unlock Exercise 3