GW150914 and the First Direct Detection

GW150914 was the opening note of an entire observational field. Advanced LIGO and its partner Advanced Virgo have since catalogued roughly a hundred compact-binary mergers, mapping the mass and spin distribution of stellar black holes — including systems in the "pair-instability" mass gap and the intermediate-mass regime — and populating the census that constrains how massive stars die and how binaries form, whether by isolated evolution or dynamical capture in dense clusters. The multi-messenger promise was realized on 17 August 2017 with GW170817, a binary-neutron-star merger seen in gravitational waves and, $1.7\,\mathrm{s}$ later, in gamma rays, then across the electromagnetic spectrum as a kilonova — pinning down the origin of heavy elements and delivering a standard-siren measurement of the Hubble constant independent of the cosmic distance ladder. The ringdown of ever-louder events is now a working laboratory for the black-hole no-hair theorem, testing whether the remnant's quasinormal spectrum is fixed by mass and spin alone. Looking ahead, the space-based LISA interferometer (planned for the 2030s) will open the millihertz band to supermassive-black-hole mergers and galactic binaries, while pulsar-timing arrays have already detected the nanohertz gravitational-wave background from the population of supermassive binaries across the cosmos — the same physics Weiss budgeted in 1972, now spanning nine decades of frequency.

Two black holes in a bound orbit are a time-varying mass quadrupole, and by the quadrupole formula they radiate gravitational waves. Radiation carries energy away, so the orbit shrinks; a tighter orbit is a faster orbit (Kepler), and a faster orbit radiates at a higher frequency and larger amplitude. Frequency and amplitude therefore climb together in a runaway — the "chirp" — until the two horizons touch and merge. What survives is a single, distorted, spinning Kerr black hole that sheds its lumps by ringing like a struck bell: a few damped oscillations, the ringdown, whose pitch and decay are fixed by the remnant's mass and spin alone. The first figure is that story drawn as a strain-versus-time curve, the actual shape LIGO recorded.

An interferometer turns that faint stretch into a phase shift. A plus-polarized wave arriving down the beam splitter lengthens one arm while shortening the perpendicular one, so the differential arm length swings by $\Delta L \sim h L$. With $L = 4\,\mathrm{km}$ and $h\sim10^{-21}$ that is only $\sim 4\times10^{-18}\,\mathrm{m}$, a thousandth of a proton radius — but the two beams recombine so that this difference in light travel-time shows up as a change in the interference fringe, amplified by hundreds of round trips in the resonant arms and by tens of kilowatts of circulating laser power. The second figure shows why a gravitational-wave detector is fundamentally a comparator of two orthogonal light paths, not a ruler laid against a fixed lattice.

The physics is stitched from three regimes that never before had to agree. The early inspiral is analytic — the post-Newtonian expansion and the quadrupole formula handle $v/c\lesssim 0.3$. The merger, where $v/c\to 0.5$ and the field is fully nonlinear, yields only to numerical relativity solving $G_{\mu\nu}=8\pi T_{\mu\nu}$ on a supercomputer. The ringdown is linear perturbation theory of a Kerr black hole, a superposition of quasinormal modes. That all three phases splice into one continuous waveform matching the data — with the remnant mass and spin from the merger predicting the ringdown pitch measured independently — is the strong-field test of general relativity that a century of theory had been waiting to take.

The observable is a dimensionless strain: the fractional change in proper separation between two freely-falling test masses as the wave passes. For LIGO those masses are the suspended mirrors, and the strain is read as a differential arm length.

where: $h$ — gravitational-wave strain amplitude, dimensionless; $\Delta L$ — differential arm-length change; $L=4\,\mathrm{km}$ — LIGO arm length. Reading: the peak strain of GW150914 was one part in $10^{21}$, so $\Delta L \approx 4\times10^{-18}\,\mathrm{m}$ — the entire detection is a mirror moving by a thousandth of a proton radius. The inspiral is governed almost entirely by a single combination of the two masses, the chirp mass. It is the mass parameter that controls how fast the frequency sweeps, and it is what the long, many-cycle inspiral measures most tightly.

where: $m_1,m_2$ — the two component masses ($36\,M_\odot$ and $29\,M_\odot$, source frame); $\mathcal{M}$ — chirp mass, $\approx 28\,M_\odot$ (source frame). Reading: two very different mass pairs with the same $\mathcal{M}$ give nearly identical early inspirals; breaking that degeneracy to get $m_1,m_2$ separately needs the higher-frequency, higher-order part of the signal. The frequency evolution follows from equating the quadrupole luminosity to the rate of orbital energy loss. At leading (Newtonian, quadrupole) order the result depends only on the chirp mass — the origin of its name and of its precise measurability.

where: $f$ — gravitational-wave frequency, twice the orbital frequency; $\dot f$ — its time derivative, the chirp rate. Reading: measure $f$ and $\dot f$ at any instant and equation (3) hands you $\mathcal{M}$ with no other input — a self-calibrating "standard siren." Inverting (3) turns two directly observed numbers into the chirp mass — the back-of-envelope measurement LIGO used to argue the source was a stellar-mass binary before any template fitting.

where: all symbols as above. Reading: near $f\approx 150\,\mathrm{Hz}$ the observed chirp gives $\mathcal{M}\approx 30\,M_\odot$ in the detector frame, $\approx 28\,M_\odot$ after correcting for the cosmological redshift $z\approx 0.09$. The inspiral ends when the orbit reaches the innermost stable circular orbit (ISCO) of the total mass; beyond it no stable orbit exists and the bodies plunge. The ISCO frequency scales inversely with total mass and sets where inspiral hands off to merger.

where: $M=m_1+m_2$ — total mass; the numerical form uses the Schwarzschild test-particle ISCO. Reading: for $M\approx 65\,M_\odot$ this gives $f_{\rm ISCO}\approx 68\,\mathrm{Hz}$, so the low-frequency band is a clean inspiral and everything above $\sim 100\,\mathrm{Hz}$ is the strong-field merger and ringdown. Energy conservation fixes the total radiated energy as the mass deficit between the two parents and the single remnant, and its time derivative peaks at merger with a power that is astonishing in absolute terms.

where: $M_f=62^{+4}_{-4}\,M_\odot$ — final black-hole mass; $E_{\rm rad}$ — energy carried off by gravitational waves, about $5\%$ of the system's rest mass; $L_{\rm peak}$ — peak luminosity, $\approx 200\,M_\odot c^2\,\mathrm{s}^{-1}$. Reading: for a few milliseconds the binary out-radiated, in gravitational waves, the entire electromagnetic output of the observable universe. Finally, what a given detector actually records is the two wave polarizations projected through its direction-dependent antenna pattern — the reason a single L-shaped instrument cannot, by itself, locate a source on the sky.

where: $F_+,F_\times$ — antenna-pattern functions; $h_+,h_\times$ — the transverse-traceless plus and cross polarization amplitudes; $\theta,\phi,\psi$ — source sky position and polarization angle. Reading: two separated detectors are needed to triangulate from the time delay — GW150914's $6.9\,\mathrm{ms}$ Hanford–Livingston lag confined it to a $\sim 600\,\deg^2$ arc of the southern sky.

The analysis of a single event spans every regime of the theory, from slow-motion weak field to full nonlinearity to linear perturbation theory — and each regime is owned by a different piece of the toolkit.

1 · Chirp mass and chirp rate (fully worked)

From the reported source-frame masses $m_1=36\,M_\odot$, $m_2=29\,M_\odot$, compute the chirp mass, then use the chirp equation to predict $\dot f$ at $f=35\,\mathrm{Hz}$ and $f=150\,\mathrm{Hz}$ and check that the signal must sweep through the band in a fraction of a second.

Step 1. Chirp mass from equation (2): the product $m_1m_2 = 1044\,M_\odot^2$, the sum $m_1+m_2=65\,M_\odot$, so $\mathcal{M}=(1044)^{3/5}/(65)^{1/5}\,M_\odot = 64.8/2.30\,M_\odot = \mathbf{28.1\,M_\odot}$. This is the source-frame value; the observed (detector-frame) chirp mass is larger by $(1+z)\approx 1.09$, about $30.6\,M_\odot$.

Step 2. Put $\mathcal{M}$ into geometric units: $G\mathcal{M}/c^3 = 28.1\times 4.925\times10^{-6}\,\mathrm{s} = 1.38\times10^{-4}\,\mathrm{s}$. Raise to the $5/3$ power: $(1.38\times10^{-4})^{5/3}=2.9\times10^{-6}\,\mathrm{s}^{5/3}$.

Step 3. Evaluate the chirp rate (3) at $f=35\,\mathrm{Hz}$: $\dot f = \tfrac{96}{5}\pi^{8/3}(2.9\times10^{-6})(35)^{11/3}$. With $\pi^{8/3}=21.0$ and $35^{11/3}=1.0\times10^{6}$, this gives $\dot f \approx \mathbf{69\,\mathrm{Hz/s}}$ — at the bottom of the band the frequency is climbing modestly.

Step 4. Repeat at $f=150\,\mathrm{Hz}$: $150^{11/3}=1.5\times10^{8}$, so $\dot f \approx \mathbf{1.4\times10^{4}\,\mathrm{Hz/s}}$. The chirp rate has grown two hundredfold, exactly the runaway the $f^{11/3}$ scaling predicts. Integrating, the signal spends most of its band-time near the low-frequency end and races through the last decade of frequency in tens of milliseconds — the whole audible-band signal lasts only about 0.2 s, precisely the observed chirp.

2 · The energy budget (fully worked)

Verify that $\sim 3\,M_\odot c^2$ was radiated and put the peak luminosity in perspective.

Step 1. Mass deficit: $m_1+m_2 = 65\,M_\odot$, remnant $M_f = 62\,M_\odot$, so $E_{\rm rad} = (65-62)\,M_\odot c^2 = 3\,M_\odot c^2$. In joules, $E_{\rm rad}=3\times(1.99\times10^{30}\,\mathrm{kg})\times(3.0\times10^8\,\mathrm{m/s})^2 = \mathbf{5.4\times10^{47}\,\mathrm{J}}$.

Step 2. For scale, the Sun radiates $\sim 3.8\times10^{26}\,\mathrm{W}$; over its entire $\sim 10^{10}$-year main-sequence life it emits $\sim 1.2\times10^{44}\,\mathrm{J}$. GW150914 released about four thousand Suns' worth of lifetime energy in two-tenths of a second — as spacetime ripples, not light.

Step 3. Peak luminosity: $L_{\rm peak}\approx 3.6\times10^{49}\,\mathrm{W}$. Dividing by $M_\odot c^2 = 1.8\times10^{47}\,\mathrm{J}$ gives $\approx \mathbf{200\,M_\odot c^2\,\mathrm{s}^{-1}}$. In solar luminosities that is $\sim 10^{23}\,L_\odot$ — comparable to the combined light of all $\sim 10^{22}$–$10^{23}$ stars in the observable universe, for a few milliseconds.

Step 4. Reconcile that with the tiny strain at Earth. Amplitude falls as $1/r$ and flux as $1/r^2$; at $r\approx 410\,\mathrm{Mpc}=1.3\times10^{25}\,\mathrm{m}$ even this colossal luminosity is diluted to $h\sim 10^{-21}$. The faintness is distance and the $G/c^4$ coupling, not weak emission.

3 · From strain to mirror displacement

What physical displacement does $h\sim 1.0\times10^{-21}$ produce, and how does it compare to a proton?

Step 1. Differential arm length from equation (1): $\Delta L = hL = (1.0\times10^{-21})(4\times10^{3}\,\mathrm{m}) = \mathbf{4\times10^{-18}\,\mathrm{m}}$.

Step 2. Compare to a proton radius, $r_p\approx 0.84\times10^{-15}\,\mathrm{m}$: $\Delta L/r_p \approx 4\times10^{-18}/8.4\times10^{-16} \approx \mathbf{5\times10^{-3}}$, i.e. about one-thousandth of a proton radius. Achieving this requires resonant arm cavities (hundreds of round trips), tens of kilowatts of circulating power to beat photon shot noise, and seismic isolation stacks — the reason a $10^{-21}$ strain is at the very edge of what is measurable, and why it took forty years.

Gravitational-wave detection is the work of many hands over a century — Einstein's prediction, Weber's first bold attempts, the Hulse–Taylor indirect proof, the numerical-relativity breakthroughs of 2005 — but the interferometric detector that actually caught GW150914 has one intellectual father above all.

Texts. Misner, Thorne & Wheeler, Gravitation, Ch. 35–37 (gravitational waves and their generation) · Wald, General Relativity, §4.4 · Carroll, Spacetime and Geometry, Ch. 7 · Hartle, Gravity, Ch. 16 · Schutz, A First Course in General Relativity, Ch. 9 · Maggiore, Gravitational Waves, Vol. 1, Ch. 4 (chirp, matched filtering).

Provenance. All measured quantities — component masses $36$ and $29\,M_\odot$, remnant $62\,M_\odot$, radiated energy $\sim 3\,M_\odot c^2$, peak luminosity $3.6\times10^{49}\,\mathrm{W}$, peak strain $1.0\times10^{-21}$, luminosity distance $410\,\mathrm{Mpc}$ ($z\approx 0.09$), band $35$–$250\,\mathrm{Hz}$, network SNR $\approx 24$, and the $6.9\,\mathrm{ms}$ inter-site delay — are from Abbott et al. (LIGO/Virgo), "Observation of Gravitational Waves from a Binary Black Hole Merger," Phys. Rev. Lett. 116, 061102 (2016). A verifier should recompute: the chirp mass $\mathcal{M}=28.1\,M_\odot$ from $m_1,m_2$; the chirp equation and its inversion (Carroll Ch. 7, Schutz Ch. 9, Maggiore Vol. 1); the ISCO frequency $f=c^3/(6^{3/2}\pi GM)\approx 68\,\mathrm{Hz}$ for $M=65\,M_\odot$ (Schwarzschild geometry); the Kepler separation $\approx 350\,\mathrm{km}$ at $f_{\rm GW}=150\,\mathrm{Hz}$; the energy budget $5.4\times10^{47}\,\mathrm{J}$ and $L_{\rm peak}\approx 200\,M_\odot c^2\,\mathrm{s}^{-1}$; the displacement $\Delta L\approx 4\times10^{-18}\,\mathrm{m}$. The matched-filter formalism is standard (Finn 1992). The 2017 Nobel Prize (Weiss, Barish, Thorne) is historical fact.

Prerequisites. Quadrupole Formula · Transverse-Traceless Gauge · Plus and Cross Polarizations · Geodesic Deviation and the Tidal Action of Gravitational Waves · Isaacson Stress-Energy of Gravitational Waves. Leads to. Multipole Expansion of Gravitational Radiation · Helicity and the Spin-2 Nature of the Graviton.

Ask the tutor: Ask me to walk through the chirp-mass inversion on the real GW150914 numbers, or to show how the ringdown frequency tests the black-hole no-hair theorem.