Linearized Gravity and Wave Equation
| Starting from the Einstein field equations with small perturbation $g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu}$, $ | h_{\mu\nu} | \ll 1$, expand to first order. Define the trace-reversed perturbation: |
In the Lorenz gauge $\partial^\mu \bar{h}_{\mu\nu} = 0$, the linearized EFE become a wave equation:
\[\Box\, \bar{h}_{\mu\nu} = -\frac{16\pi G}{c^4} T_{\mu\nu}\]with $\Box = \eta^{\mu\nu}\partial_\mu \partial_\nu = -c^{-2}\partial_t^2 + \nabla^2$. In vacuum ($T_{\mu\nu} = 0$), this becomes the massless wave equation with solutions propagating at $c$.
The transverse-traceless (TT) gauge further simplifies the physical description. Imposing $h^\mu_{\ \mu} = 0$ (traceless) and $h_{\mu 0} = 0$, $\partial^j h_{ij} = 0$ (transverse), for a wave propagating in the $z$-direction:
\[h_{ij}^\text{TT} = \begin{pmatrix} h_+ & h_\times & 0 \\ h_\times & -h_+ & 0 \\ 0 & 0 & 0 \end{pmatrix} e^{i(kz - \omega t)}\]The two independent polarizations $h_+$ and $h_\times$ are the physical degrees of freedom. A ring of test masses is deformed into an ellipse that oscillates between stretched and compressed configurations in the $x$-$y$ plane, rotated $45°$ between polarizations.
The Two Polarizations
Plus polarization $h_+$: separation between test masses changes as
\[\delta x(t) = \frac{1}{2} h_+(t)\, x_0, \quad \delta y(t) = -\frac{1}{2} h_+(t)\, y_0\]Cross polarization $h_\times$: causes shearing
\[\delta x(t) = \frac{1}{2} h_\times(t)\, y_0, \quad \delta y(t) = \frac{1}{2} h_\times(t)\, x_0\]The strain amplitude reaching Earth from a source at distance $r$ is extraordinarily small. For GW150914 (two ~$30 M_\odot$ black holes at $\sim 400\ \text{Mpc}$):
\[h \sim 10^{-21}\]This corresponds to a displacement of $\sim 10^{-18}\ \text{m}$ over a 4 km LIGO arm—a thousandth the diameter of a proton.
Quadrupole Formula and Radiated Power
The quadrupole formula gives the leading-order gravitational wave emission from a source with time-varying mass quadrupole moment $Q_{ij}$:
\[h_{ij}^\text{TT} = \frac{2G}{c^4 r}\ddot{Q}_{ij}^\text{TT}(t - r/c)\]where the reduced quadrupole moment tensor is
\[Q_{ij} = \int \rho \left(x_i x_j - \frac{1}{3}\delta_{ij} r^2\right) d^3x\]The total radiated power (luminosity) is the Peters formula:
\[P = \frac{G}{5c^5}\left\langle \dddot{Q}_{ij}\dddot{Q}^{ij}\right\rangle = \frac{32G^4 m_1^2 m_2^2(m_1+m_2)}{5c^5 a^5(1-e^2)^{7/2}}f(e)\]for a Keplerian binary with semi-major axis $a$, eccentricity $e$, and
\[f(e) = 1 + \frac{73}{24}e^2 + \frac{37}{96}e^4\]This result, accurate to leading post-Newtonian order, matched the Hulse-Taylor pulsar orbital decay to $0.2\%$.
Binary Inspiral: Chirp Mass and Frequency Evolution
For a compact binary (black holes or neutron stars), energy loss to gravitational waves drives inspiral: the orbit shrinks while the frequency sweeps upward—a “chirp.”
The chirp mass $\mathcal{M}$ governs the frequency evolution and is the most precisely measured parameter from a GW signal:
\[\mathcal{M} = \frac{(m_1 m_2)^{3/5}}{(m_1 + m_2)^{1/5}}\]The orbital frequency evolution at leading order:
\[\dot{f} = \frac{96}{5} \pi^{8/3} \left(\frac{G\mathcal{M}}{c^3}\right)^{5/3} f^{11/3}\]This gives the time-to-merger from GW frequency $f_\text{gw} = 2 f_\text{orb}$:
\[t_\text{merge} = \frac{5c^5}{256\pi^{8/3}} \left(\frac{G\mathcal{M}}{c^3}\right)^{-5/3} f_\text{gw}^{-8/3}\]The GW frequency at merger (innermost stable circular orbit) is approximately
\[f_\text{ISCO} = \frac{c^3}{6^{3/2}\pi G M_\text{tot}} \approx \frac{4400\ \text{Hz}}{M_\text{tot}/M_\odot}\]For a $60 M_\odot$ total mass system (like GW150914), $f_\text{ISCO} \approx 150\ \text{Hz}$.
| Event | Component masses ($M_\odot$) | $\mathcal{M}$ ($M_\odot$) | Distance (Mpc) | $f_\text{peak}$ (Hz) |
|---|---|---|---|---|
| GW150914 | $35.6 + 30.6$ | 28.3 | 440 | ~150 |
| GW170817 | $1.17 + 1.36$ | 1.19 | 40 | ~1000 |
| GW190521 | $95 + 69$ | 65 | 5300 | ~60 |
LIGO Detector: Interferometry
LIGO is a Michelson interferometer with 4 km Fabry-Pérot arm cavities. A passing GW with strain $h_+$ differentially changes the arm lengths:
\[\Delta L_x = +\frac{1}{2}h_+ L, \quad \Delta L_y = -\frac{1}{2}h_+ L\]producing a differential phase shift $\Delta\phi = 4\pi\Delta L/\lambda$. Optical power recycling and signal recycling cavities store $\sim 100\ \text{kW}$ in the arms, enhancing the effective path length by a factor $\mathcal{F} \approx 300$ (finesse).
The noise power spectral density $S_n(f)$ sets the sensitivity floor. Key noise sources:
| Frequency | Dominant noise |
|---|---|
| $< 10\ \text{Hz}$ | Seismic noise |
| $10\text{–}100\ \text{Hz}$ | Suspension thermal noise |
| $\sim 100\ \text{Hz}$ | Quantum radiation pressure |
| $> 100\ \text{Hz}$ | Shot noise ($\propto 1/\sqrt{P}$) |
The shot noise arises from photon counting statistics: $S_n^\text{shot}(f) = \hbar \lambda c / (2\pi P)$. Advanced LIGO uses squeezed light to reduce shot noise below the standard quantum limit.
Matched Filtering and Detection Statistics
Gravitational wave detection is a signal-processing problem in colored noise. The optimal matched filter for a template waveform $h(t)$ embedded in noise $n(t)$ with PSD $S_n(f)$ is
\[\rho = \frac{4\,\text{Re}\int_0^\infty \frac{\tilde{d}(f)\tilde{h}^*(f)}{S_n(f)}\, df}{\sqrt{4\int_0^\infty \frac{|\tilde{h}(f)|^2}{S_n(f)} df}}\]The signal-to-noise ratio (SNR) $\rho$ is compared against a threshold (typically $\rho > 8$ for individual detectors). The detection statistic follows a noncentral chi-squared distribution.
The inner product on waveform space:
\[\langle a \mid b \rangle = 4\,\text{Re}\int_0^\infty \frac{\tilde{a}(f)\tilde{b}^*(f)}{S_n(f)}\, df\]defines an effective distance $D_\text{eff}$ that includes antenna pattern functions:
\[D_\text{eff} = D_L \left[F_+^2\left(\frac{1+\cos^2\iota}{2}\right)^2 + F_\times^2 \cos^2\iota\right]^{-1/2}\]where $F_+, F_\times$ are the antenna patterns and $\iota$ is the inclination angle.
The template bank covers the chirp mass $\mathcal{M}$ and mass ratio $q = m_2/m_1 \in [1/18, 1]$, with bank spacing chosen so the worst-case fitting factor (match between signal and nearest template) exceeds $0.97$.
Multimessenger Astronomy
GW170817 (two neutron stars merging) was detected simultaneously in gravitational waves by LIGO/Virgo and in gamma rays by Fermi-GBM $1.7\ \text{s}$ later, launching multimessenger astronomy. This single event:
- Confirmed short gamma-ray bursts originate from neutron star mergers
- Observed the kilonova (r-process nucleosynthesis) in optical/IR for 2 weeks
- Measured $H_0 = 70^{+12}_{-8}\ \text{km s}^{-1}\text{Mpc}^{-1}$ independently
- Constrained the neutron star equation of state via tidal deformability $\Lambda$
The tidal deformability parameter $\Lambda = \frac{2}{3}k_2 (c^2 R / G M)^5$ modifies the GW phasing at 5PN order and encodes neutron star interior physics. GW170817 constrained $\Lambda_{1.4} < 800$, ruling out the stiffest equations of state.
Future space-based detectors (LISA, $2037$) will observe millihertz GWs: supermassive black hole mergers ($10^6$–$10^9 M_\odot$), extreme mass-ratio inspirals (EMRIs), and the stochastic GW background from the early universe.