Modern Physics · Advanced Topics

Gravitational Waves

Ripples in spacetime curvature propagating at c — predicted by Einstein in 1916, first directly detected by LIGO in 2015. Gravitational wave astronomy has opened a new observational window on black holes, neutron stars, and the Big Bang.

PrerequisitesGeneralrelativity(Ch.GR)⋅Tensorcalculus(Ch.TC)⋅Electromagneticwaves(Ch.EM−wavGeneral relativity (Ch. GR) \cdot Tensor calculus (Ch. TC) \cdot Electromagnetic waves (Ch. EM-waves)⋅Specialrelativity(Ch.SRes) \cdot Special relativity (Ch. SR
Learning Goals
  • Derive the linearized Einstein equations in Lorenz gauge and identify the two GW polarizations.
  • Apply the quadrupole formula to compute GW strain amplitude and radiated power for a binary system.
  • Extract the chirp mass from an observed df/dt and use it to estimate the source distance.
  • Explain the noise sources limiting LIGO sensitivity and how squeezed light surpasses the SQL.
  • Describemulti−messengerastronomywithGW170817andhowstandardsirensconstrainH0Describe multi-messenger astronomy with GW170817 and how standard sirens constrain H_{0}.

GW.1 Linearized General Relativity

In the weak-field limit, write g_μν = η_μν + h_μν with |h_μν| ≪ 1. The Einstein equations linearize. In Lorenz (harmonic) gauge ∂^μ h̄_μν = 0 (where h̄_μν = h_μν − ½η_μν h is the trace-reversed perturbation):

□hˉμν=−16\piG/c4Tμν(linearizedEinsteinequations)\Box h̄_\mu\nu = -16\piG/c^{4} T_\mu\nu \qquad (linearized Einstein equations)(GW.1)

In vacuum (T_μν = 0), this is a wave equation: □ h̄_μν = 0, giving waves propagating at speed c. Residual gauge freedom → transverse-traceless (TT) gauge: h̄_TT has only spatial components, h_ii = 0 (traceless), and h_μi ∝ k^i = 0 (transverse). Two independent polarizations: plus (+) and cross (×).

Definition GW.1 — GW Polarizations
A GW propagating in the z-direction in TT gauge: h_\mu\nu = h_+ e_+^{\mu\nu} cos(kz - \omegat) + h_\times e_×μνcos(kz−\omegat+ϕ)wheree+=diag(0,1,−1,0)/2ande×hasoff−diagonal±1/2inx\times^{\mu\nu} cos(kz - \omegat + \phi) where e_+ = diag(0, 1, -1, 0)/2 and e_\times has off-diagonal \pm1/2 in xy. The plus polarization stretches x while compressing y (and vice versa), oscillating at 2f_ The cross polarization does the same but rotated by 45°.

GW.2 Generation: The Quadrupole Formula

GWs are produced by changing quadrupole moments (not monopole/dipole — those are conserved by mass/momentum conservation). The leading-order emission:

h+,h× (2G)/(rc4)×d2Iij/dt2(quadrupoleformula,Iij=reducedquadrupolemoment)h_+, h_\times ~ (2G)/(r c^{4}) \times d^{2}I_{ij}/dt^{2} \qquad (quadrupole formula, I_{ij} = reduced quadrupole moment)(GW.2)
PGW=G/(5c5)×⟨\cdotsIij\cdotsIij⟩(radiatedpower,⋯=thirdtimederivative)P_{GW} = G/(5c^{5}) \times ⟨\cdotsI_ij \cdotsI_ij⟩ \qquad (radiated power, \cdots = third time derivative)(GW.3)

The factor G/c⁵ = 3.6×10⁻⁵³ W⁻¹ makes GW emission extremely weak for laboratory sources. Only compact astrophysical objects (NS, BH) emit detectable GWs.

Theorem GW.1 — Inspiral and Merger
Abinarysystemofmassesm1,m2(chirpmassMc=(m1m2)3/5/(m1+m2)1/5)losesenergA binary system of masses m_{1}, m_{2} (chirp mass M_{c} = (m_{1}m_{2})^{3/5}/(m_{1}+m_{2})^{1/5}) loses energytoGWsandspiralsinward.Thefrequencyevolvesas:df/dt=(96/5)π8/3(GMc/c3)(5/3y to GWs and spirals inward. The frequency evolves as: df/dt = (96/5)\pi^{8/3}(GM_c/c^{3})^(5/3) f11/3Thestrainamplitude:h (4/r)(GMc/c2)5/3(\pif/c)2/3Atmerger(Schwarzschif^{11/3} The strain amplitude: h ~ (4/r)(GM_c/c^{2})^{5/3}(\pif/c)^{2/3} At merger (Schwarzschi radiuscontact):fISCO≈c3/(63/2πGMtotal)≈4400Hz×(M⊙/Mtotalradius contact): f_{ISCO} \approx c^{3}/(6^{3/2} \pi G M_{total}) \approx 4400 Hz \times (M_\odot/M_{total}
Example GW.1 — GW150914 — First Direct Detection

LIGO detected GW150914 onSeptember14,2015.Reconstructkeyparametersfromtheobservedsignal:fsweeps35→on September 14, 2015. Reconstruct key parameters from the observed signal: f sweeps 35 \to strainh≈10−21,estimateddistance 410Mpcstrain h \approx 10^{-21}, estimated distance ~410 Mpc

Chirp mass from df/dt:Atf≈100Hz,df/dt≈(150−35)/0.2Hz/s≈575Hz/s.M=(c3/(Gπ8/3))×(5/96)×(df/dAt f \approx 100 Hz, df/dt \approx (150-35)/0.2 Hz/s \approx 575 Hz/s. M = (c^{3}/(G \pi^{8/3})) \times (5/96) \times (df/dt)3/5×f−11/5≈28.3M⊙.Thisisthebest−measuredparameterfromtheinspiralphat)^{3/5} \times f^{-11/5} \approx 28.3 M_\odot. This is the best-measured parameter from the inspiral phase (fractional uncertainty ~1%).
Mass scale from high-frequency cutoff:The observed high-frequency merger/ringdown near 150 Hz is not a clean Schwarzschild ISCO frequency, but it sets thecompact−objectmassscale.Detailedwaveformmatchinggivesinitialblack−holemasses≈he compact-object mass scale. Detailed waveform matching gives initial black-hole masses \approx048W—brieflyoutshiningallstarsintheobservableuniversecombined0^{48} W — briefly outshining all stars in the observable universe combined.
Distance from strain:h (4G5/3(\pifM)2/3)/(c4r).Atf=100Hz,M=28M⊙:h≈10−21→r≈410Mpc.LIh ~ (4 G^{5/3} (\pifM)^{2/3})/(c^{4} r). At f = 100 Hz, M = 28 M_\odot: h \approx 10^{-21} \to r \approx 410 Mpc. LIGOarmlengthL=4km;lengthchange\DeltaL=hL/2≈10−21×4000m/2=2×10−18m=2am(attGO arm length L = 4 km; length change \DeltaL = h L/2 \approx 10^{-21} \times 4000m/2 = 2\times10^{-18} m = 2 am (attometers). 1/500 of a proton radius.
Significance:Signal−to−noiseratioSNR≈24.Detectionsignificance:5.3σ(falsealarmrate<1per200Signal-to-noise ratio SNR \approx 24. Detection significance: 5.3\sigma (false alarm rate < 1 per 200,000 yr). Confirmed as two black holes — first direct observation ofabinaryblackholemerger,andfirstobservationofblackholeswithmassesM>20M⊙of a binary black hole merger, and first observation of black holes with masses M > 20 M_\odot

GW.3 LIGO Interferometer

LIGO uses a modified Michelson interferometer with 4 km arms. A GW stretches one arm and squeezes the other: ΔL = h L/2.

Noise sources (from high to low frequency): 1. Seismic noise (below ~10 Hz): ground vibrations. Mitigated by 4-stage pendulum isolation. 2. Thermal noise (10–200 Hz): Brownian motion of mirror coatings and suspensions. Fused silica fibers, low-loss coatings (SiO₂/Ta₂O₅), low temperature prototypes. 3. Quantum noise (above ~100 Hz): photon shot noise. Standard quantum limit: h_SQL = (1/L)√(8ℏ/(mω²)) — trades off shot noise against radiation pressure noise. Overcome by squeezed light injection (LIGO O3: 15 dB squeezing applied).

Power recycling: mirror between laser and BS reflects light back → builds up 200 kW of intracavity power (from 20 W laser input). Signal recycling: mirror at dark port tunes the detector's frequency response. Advanced LIGO sensitivity (O4): h ~ 3×10⁻²⁴/√Hz at 100 Hz, horizon distance ~200 Mpc for binary neutron star mergers.

GW.4 Sources and Multi-Messenger Astronomy

Binary black holes (BBH): most numerous detections (~100+ by O3 end). No EM counterpart (BH mergers don't produce photons).Binary neutron stars (BNS): GW170817 (2017) — first BNS detection. Simultaneous gamma-ray burst GRB170817A detected 1.7 s after merger by Fermi. Multi-messenger observation confirmed: NS mergers = short GRBs + kilonovae (r-process nucleosynthesis — gold, platinum produced).

Hubble constant from GW: "standard siren" — GW gives absolute distance (no distance ladder), EM gives redshift. GW170817: H₀ = 70⁺¹²_(-8) km/s/Mpc. With more events: will resolve Hubble tension model-independently.

Pulsar timing arrays (PTAs): millisecond pulsars as a GW detector. Nanohertz GW background (f ~ 1–100 nHz) from supermassive black hole binaries. First evidence (NANOGrav 2023, ~5σ) for GW background — new frequency window.LISA (2030s): space-based, 2.5 Mkm arms, millihertz band — targets SMBH mergers, extreme mass-ratio inspirals, stochastic GW background from inflation.

Definition GW.2 — Common Traps
  • Gravitational waves are strain waves: detectors measure fractional length change, not a force meter reading.
  • Monopole and dipole radiation are absent: gravitational radiation begins at quadrupole order.
  • Polarizations are transverse tidal patterns: plus and cross describe stretching directions.
  • Signal frequency tracks orbital motion: inspiral chirps upward as the orbit shrinks.
Exercises — GW.1–GW.4 Gravitational Waves
1.Describe how the plus polarization of a gravitational wave deforms a ring of test masses. What is the antenna pattern F_er?
Straightforward
2.
Derive the gravitational wave luminosity (Peters formula) for an equal-mass circular binary. Compute the merger timescale for the Hulse-Taylor pulsar.
Myr
Intermediate
3.
Explainhowbinaryneutronstarmergersserveas"standardsirens"formeasuringH0.UseGExplain how binary neutron star mergers serve as "standard sirens" for measuring H_{0}. Use GW170817data(DL=40Mpc,z=0.0099)toestimateH0andcomparetoothermeasurementsW170817 data (D_{L} = 40 Mpc, z = 0.0099) to estimate H_{0} and compare to other measurements.
km/s/Mpc
Intermediate
4.Derive the standard quantum limit (SQL) for a gravitational wave detector by balancing shot noise and radiation pressure noise. How does squeezed light injection help LIGO surpass the SQL?
Challenging
Key Takeaways
  • Linearized GR: \Boxh̄_\mu\nu = -16\piG/c^{4} T_\mu\nu. TT gauge: two polarizations h_+, h_\times propagating at c.
  • Quadrupoleformula:h (2G/rc4)×d2I/dt2.PowerP (G/5c5)(\cdotsI)2—extremelyweakforQuadrupole formula: h ~ (2G/rc^{4}) \times d^{2}I/dt^{2}. Power P ~ (G/5c^{5})(\cdotsI)^{2} — extremely weak for lab sources.
  • Inspiral:chirpmassMfromdf/dt.Strainh (GM)5/3/(c4r).MergeratfISCO≈4400HInspiral: chirp mass M from df/dt. Strain h ~ (GM)^{5/3}/(c^{4} r). Merger at f_{ISCO} \approx 4400 Hz×M⊙/Mtotalz \times M_\odot/M_{total}.
  • GW150914:BBHmerger,3M\odotc2radiated,\DeltaL 2am(2×10−18m).5.3σ.Nobel2017GW150914: BBH merger, 3 M_\odotc^{2} radiated, \DeltaL ~ 2 am (2\times10^{-18} m). 5.3\sigma. Nobel 2017.
  • LIGO noise: seismic(<10Hz),thermal(10−200Hz),quantum(shot+radiationpressure).SQL 3.7×1seismic (<10 Hz), thermal (10-200 Hz), quantum (shot + radiation pressure). SQL ~ 3.7\times1
  • GW170817(BNS):confirmedNSmergers=shortGRBs+kilonovae.H0=70km/s/Mpcas"standaGW170817 (BNS): confirmed NS mergers = short GRBs + kilonovae. H_{0} = 70 km/s/Mpc as "standard siren".