Gravitational Waves

Introduction

Gravitational waves are ripples in spacetime itself, predicted by Einstein's general relativity in 1916 and directly detected by LIGO in 2015. These disturbances propagate at the speed of light, carrying information about the most violent events in the universe: colliding black holes, merging neutron stars, supernovae, and the Big Bang itself.

1. Linearized Gravity

Metric Perturbation

Consider weak gravitational fields as perturbations around flat spacetime:

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

Einstein Equations to First Order

Working to linear order in $h_{\mu\nu}$, the Einstein equations become:

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

where $\Box = \eta^{\mu\nu}\partial_\mu\partial_\nu$ is the d'Alembertian and $\bar{h}_{\mu\nu}$is the trace-reversed perturbation:

$$\bar{h}_{\mu\nu} = h_{\mu\nu} - \frac{1}{2}\eta_{\mu\nu}h, \quad h = \eta^{\mu\nu}h_{\mu\nu}$$

Gauge Freedom

Under infinitesimal coordinate transformations $x^\mu \to x^\mu + \xi^\mu$:

$$h_{\mu\nu} \to h_{\mu\nu} - \partial_\mu\xi_\nu - \partial_\nu\xi_\mu$$

We can impose the Lorenz gauge:

$$\partial^\mu\bar{h}_{\mu\nu} = 0$$

Vacuum Wave Equation

In vacuum ($T_{\mu\nu} = 0$), gravitational waves satisfy:

$$\Box \bar{h}_{\mu\nu} = 0$$

This is a wave equation with solutions propagating at speed $c$!

2. Plane Wave Solutions

Monochromatic Plane Wave

$$\bar{h}_{\mu\nu} = A_{\mu\nu}e^{ik_\rho x^\rho} = A_{\mu\nu}e^{i(kz - \omega t)}$$

with dispersion relation $\omega^2 = k^2c^2$ (massless waves).

Transverse-Traceless (TT) Gauge

Additional gauge fixing gives the TT gauge where:

  • • $h^{TT}_{0\mu} = 0$ (purely spatial)
  • • $h^{TT}_{ij}k^j = 0$ (transverse to propagation)
  • • $h^{TT}_{ii} = 0$ (traceless)

For wave propagating in $z$-direction, only two polarizations survive:

$$h^{TT}_{\mu\nu} = \begin{pmatrix} 0 & 0 & 0 & 0 \\ 0 & h_+ & h_\times & 0 \\ 0 & h_\times & -h_+ & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix}$$

Polarization States

Plus polarization (+):

$$h_+ = A_+\cos(\omega t - kz)$$

Stretches space in x-direction, compresses in y-direction.

Cross polarization (×):

$$h_\times = A_\times\cos(\omega t - kz + \phi)$$

Rotated 45° with respect to plus mode. GR predicts only these two helicity-2 polarizations.

3. Effect on Test Masses

Geodesic Deviation

The proper distance between two nearby test masses changes as:

$$\Delta L_i(t) = L_0\left[\delta_{ij} + \frac{1}{2}h^{TT}_{ij}(t)\right]\Delta x^j$$

Strain

The fractional change in length (strain) for arms along x and y axes:

$$\frac{\Delta L_x - \Delta L_y}{L_0} = h_+(t)$$

For LIGO's first detection (GW150914): $h \sim 10^{-21}$, corresponding to$\Delta L \sim 10^{-18}$ m for $L_0 = 4$ km arms!

Ring of Particles

A ring of free test particles deforms into an ellipse as the wave passes, with the major and minor axes oscillating perpendicular to each other for + and × polarizations.

4. Generation of Gravitational Waves

Quadrupole Formula

In the slow-motion, weak-field limit, the leading contribution comes from the second time derivative of the quadrupole moment:

$$h_{ij}^{TT}(t,\vec{x}) = \frac{2G}{c^4r}\ddot{Q}_{ij}^{TT}\left(t - \frac{r}{c}\right)$$

where the reduced quadrupole moment is:

$$Q_{ij} = \int \rho(t,\vec{x}')\left(x_i'x_j' - \frac{1}{3}\delta_{ij}|\vec{x}'|^2\right)d^3x'$$

Energy Radiated

The power radiated in gravitational waves:

$$P_{GW} = \frac{G}{5c^5}\left\langle\dddot{Q}_{ij}\dddot{Q}^{ij}\right\rangle$$

This is the famous quadrupole formula. Note: no monopole or dipole radiation (conservation of mass and momentum).

Binary System

For a circular binary with masses $m_1, m_2$, separation $r$, and angular frequency $\omega$:

$$P_{GW} = \frac{32}{5}\frac{G^4}{c^5}\frac{(m_1m_2)^2(m_1+m_2)}{r^5}$$

The orbital frequency increases as energy is lost:

$$\dot{f} = \frac{96}{5}\pi^{8/3}\left(\frac{GM_c}{c^3}\right)^{5/3}f^{11/3}$$

where $M_c = (m_1m_2)^{3/5}/(m_1+m_2)^{1/5}$ is the chirp mass.

5. Astrophysical Sources

Binary Black Hole Mergers

Masses: $5M_\odot - 100M_\odot$ each. GW frequency at ISCO:

$$f_{\text{ISCO}} \approx \frac{c^3}{6^{3/2}\pi GM} \approx 220\text{ Hz}\left(\frac{30M_\odot}{M}\right)$$

Waveform consists of three phases:

  • • Inspiral: Quasi-circular orbits, frequency increases (chirp)
  • • Merger: Plunge, highly nonlinear, strongest emission
  • • Ringdown: Final black hole settles to Kerr via quasinormal modes

Binary Neutron Star Mergers

Masses: $1-2M_\odot$ each. Higher frequency than BBH. GW170817: first NS-NS detection, accompanied by electromagnetic counterparts (kilonova, gamma-ray burst).

Supernovae and Core Collapse

Produce GW burst if asymmetric. Expected strain:

$$h \sim 10^{-21}\left(\frac{\epsilon}{10^{-6}}\right)\left(\frac{10\text{ kpc}}{d}\right)$$

where $\epsilon = E_{GW}/Mc^2$ is the efficiency. Not yet detected.

Continuous Waves

From rotating neutron stars with asymmetries (mountains, r-modes). Frequency:

$$f_{GW} = 2f_{\text{spin}}$$

Strain: $h \sim 10^{-26}$ for typical pulsars at 1 kpc. Long integration times compensate for weakness.

Stochastic Background

Superposition of unresolved sources. Characterized by energy density:

$$\Omega_{GW}(f) = \frac{1}{\rho_c}\frac{d\rho_{GW}}{d\ln f}$$

Sources: primordial (inflation, phase transitions, cosmic strings), astrophysical (unresolved binaries).

6. Detection Methods

Laser Interferometry

Michelson interferometer with arms of length $L$. Phase shift due to GW:

$$\Delta\Phi = \frac{2\pi}{\lambda}h(t)L$$

LIGO: $L = 4$ km, $\lambda = 1064$ nm, sensitivity $h \sim 10^{-23}$ at 100 Hz.

Ground-Based Detectors

  • • LIGO (USA): 2 detectors, 4 km arms, 10 Hz - 5 kHz
  • • Virgo (Italy): 3 km arms, similar band
  • • KAGRA (Japan): Underground, cryogenic, 3 km
  • • Einstein Telescope (proposed): 10 km, underground, 1 Hz - 10 kHz
  • • Cosmic Explorer (proposed): 40 km arms in USA

Space-Based Detectors

LISA (Laser Interferometer Space Antenna):

  • • 3 spacecraft in heliocentric orbit, $L = 2.5 \times 10^6$ km
  • • Frequency band: $10^{-4}$ Hz - 1 Hz
  • • Targets: supermassive black hole mergers, extreme mass ratio inspirals, Galactic binaries

Pulsar Timing Arrays

Monitor arrival times of pulses from millisecond pulsars. GWs cause correlated timing residuals:

$$\delta t \sim h \times t_{\text{obs}}$$

Sensitive to nHz frequencies. NANOGrav 2023: Evidence for stochastic background, likely from supermassive black hole binaries.

7. Major Detections

GW150914 (September 14, 2015)

First direct detection! Binary black hole merger:

  • • Masses: $36M_\odot + 29M_\odot \to 62M_\odot$
  • • Energy radiated: $3M_\odot c^2$
  • • Distance: 440 Mpc ($z = 0.09$)
  • • Peak strain: $h \sim 10^{-21}$
  • • Confirmed GR to 1 part in $10^{15}$

GW170817 (August 17, 2017)

Binary neutron star merger with EM counterpart:

  • • Masses: $1.46M_\odot + 1.27M_\odot$
  • • Distance: 40 Mpc
  • • GRB 170817A detected 1.7s later
  • • Kilonova AT2017gfo observed across EM spectrum
  • • Confirmed neutron star matter equation of state constraints
  • • Independent Hubble constant: $H_0 = 70^{+12}_{-8}$ km/s/Mpc
  • • Confirmed GW speed equals light speed: $|v_{GW}/c - 1| < 10^{-15}$

Catalog Summary (Through O3)

As of GWTC-3 (90 events total):

  • • ~85 binary black hole mergers
  • • 2 binary neutron star mergers
  • • 2 neutron star-black hole mergers
  • • Mass range: $1.4M_\odot$ to $150M_\odot$

8. Gravitational Wave Cosmology

Standard Sirens

GW signals provide luminosity distance $d_L$ from amplitude. With EM counterpart giving redshift $z$, measure Hubble constant:

$$H_0 = \frac{cz}{d_L(z)}$$

This is independent of cosmic distance ladder! Future: measure $H_0$ to ~1% with ~50 NS-NS mergers with EM counterparts.

Modified Gravity Tests

Test dispersion relation:

$$v_{GW}^2 = c^2\left(1 + \alpha\frac{f}{f_0}\right)$$

GR predicts $\alpha = 0$. Current limit: $|\alpha| < 10^{-15}$.

Primordial Gravitational Waves

From inflation, tensor perturbations produce stochastic background:

$$\Omega_{GW}^{\text{prim}}(f) \sim \Omega_{\gamma}r\left(\frac{g_*(T_f)}{g_*(T_0)}\right)^{1/3}$$

where $r$ is the tensor-to-scalar ratio. CMB B-modes constrain: $r < 0.06$. Direct detection would revolutionize our understanding of inflation!

9. Future Prospects

Science Goals

  • • Black hole populations: Mass distribution, formation channels, merger rates
  • • Nuclear physics: Neutron star equation of state from tidal deformability
  • • Tests of GR: Strong-field regime, black hole no-hair theorem
  • • Cosmology: $H_0$, dark energy equation of state, primordial GWs
  • • Fundamental physics: Speed of gravity, graviton mass, extra dimensions

Upcoming Milestones

  • • LIGO/Virgo O4 run (2023-2025): 100+ detections expected
  • • KAGRA at design sensitivity: Better sky localization
  • • LISA launch (~2035): Millihertz window opens
  • • Einstein Telescope & Cosmic Explorer: 3rd generation, 1000s of events
  • • PTA detection of individual SMBH binaries