Gravitational Waves

Einstein predicted in 1916 that accelerating masses should generate ripples in spacetime, gravitational waves: propagating at the speed of light. For a century, they remained theoretical. Then, on September 14, 2015, the LIGO detectors registered a signal from two black holes merging 1.3 billion light-years away. Spacetime had been observed to vibrate. In this lesson we develop the theory of gravitational waves from linearized gravity, explore their two polarizations, derive the quadrupole formula for radiation, and survey the new field of gravitational wave astronomy.

Linearized Gravity

To find wave solutions, we start with nearly flat spacetime. Write the metric as a small perturbation around the Minkowski background:

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

The Einstein equations, expanded to first order in $h_{\mu\nu}$ and ignoring higher-order terms, become linear. This is the regime of linearized gravity.

Under an infinitesimal coordinate transformation $x^\mu \to x^\mu + \xi^\mu$, the perturbation transforms as $h_{\mu\nu} \to h_{\mu\nu} - \partial_\mu \xi_\nu - \partial_\nu \xi_\mu$. This is the gravitational analogue of gauge invariance in electromagnetism. We can use this freedom to impose the Lorenz gauge (also called de Donder or harmonic gauge):

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

where $\bar{h}_{\mu\nu} = h_{\mu\nu} - \frac{1}{2}\eta_{\mu\nu} h$ is the trace-reversed perturbation ($h = \eta^{\mu\nu}h_{\mu\nu}$).

The Wave Equation

In the Lorenz gauge, the linearized Einstein equations in vacuum become:

With a source, the equation becomes $\Box \bar{h}_{\mu\nu} = -\frac{16\pi G}{c^4} T_{\mu\nu}$, whose solution involves the retarded Green's function, just like electromagnetic radiation.

Transverse-Traceless Gauge

For a plane wave propagating in the $z$-direction, we can further fix the gauge to the transverse-traceless (TT) gauge, in which:

• $h_{0\mu} = 0$ (purely spatial)
• $h_{iz} = 0$ (transverse to propagation)
• $h^i{}_i = 0$ (traceless)

The only nonzero components are $h_{xx} = -h_{yy} \equiv h_+$ and $h_{xy} = h_{yx} \equiv h_\times$. These are the two independent polarizations.

$$h^{TT}_{ij} = \begin{pmatrix} h_+ & h_\times & 0 \\ h_\times & -h_+ & 0 \\ 0 & 0 & 0 \end{pmatrix} \cos[\omega(t - z/c)]$$

Two Polarizations

A gravitational wave stretches and squeezes space in the directions transverse to its propagation. Consider a ring of free test particles in the $xy$-plane as a wave passes in the $z$-direction:

The $+$ polarization stretches along $x$ while compressing along $y$, then vice versa. The $\times$ polarization does the same but rotated by 45 degrees. This is the hallmark of a spin-2 field: the polarization pattern repeats after a 180-degree rotation (compared to 360 degrees for electromagnetic waves, which are spin-1).

The Quadrupole Formula

What generates gravitational waves? Electromagnetic radiation requires a time-varying dipole moment. Gravitational radiation requires a time-varying quadrupole moment. There is no gravitational dipole radiation because conservation of momentum prevents the mass dipole from accelerating (unlike electric charge, where positive and negative charges can oscillate against each other).

The factor $G/c^4 \approx 8.3 \times 10^{-45}$ s$^2$/(kg$\cdot$m) is extraordinarily small. This is why gravitational waves are so weak: you need enormous masses undergoing violent acceleration to produce detectable signals.

Energy Radiated

Gravitational waves carry energy away from their source. The power radiated is:

$$P = \frac{G}{5c^5} \left\langle \dddot{I}_{ij} \dddot{I}^{ij} \right\rangle$$

For a binary system of two equal masses $m$ in a circular orbit of radius $R$:

$$P = \frac{32 G^4 m^5}{5 c^5 R^5}$$

This energy loss causes the orbit to shrink, the orbital frequency to increase, and the gravitational wave amplitude and frequency to grow, the characteristic "chirp" signal. For two neutron stars, the final minutes before merger produce a dramatic frequency sweep from tens to thousands of hertz.

The Hulse-Taylor Binary Pulsar

The first indirect evidence for gravitational waves came from PSR B1913+16, a binary pulsar discovered by Hulse and Taylor in 1974. By timing the radio pulses over decades, they measured the orbital decay rate and found it matched Einstein's quadrupole formula to within 0.2%. The orbit shrinks by about 3.5 meters per year as gravitational wave energy is radiated away. This work earned the 1993 Nobel Prize in Physics.

LIGO and Direct Detection

LIGO (Laser Interferometer Gravitational-Wave Observatory) detects gravitational waves using laser interferometry. Two perpendicular arms, each 4 km long, form an L-shaped interferometer. A gravitational wave stretches one arm while compressing the other, changing the relative path length. LIGO measures length changes of order $10^{-18}$ meters, a thousandth the diameter of a proton.

The first detection, GW150914, came from two black holes of 36 and 29 solar masses merging to form a 62 solar mass black hole. The missing 3 solar masses were radiated as gravitational wave energy, briefly, during the final fraction of a second, this merger outshone the entire visible universe in power output.

Since then, LIGO and Virgo have detected dozens of events:

• Binary black hole mergers: the most common source, ranging from stellar-mass to intermediate-mass black holes
• Binary neutron star mergers: GW170817, also observed in electromagnetic radiation across the spectrum ("multi-messenger astronomy")
• Neutron star-black hole mergers: mixed systems

Gravitational Wave Astronomy

Gravitational waves provide information inaccessible to electromagnetic astronomy. They pass through matter unimpeded, carry information about the strongest gravitational fields, and directly probe the dynamics of spacetime itself.

Future detectors will dramatically expand our reach:

• LISA (Laser Interferometer Space Antenna): a space-based interferometer with million-kilometer arms, sensitive to millihertz frequencies from supermassive black hole mergers
• Pulsar Timing Arrays: using the precise timing of millisecond pulsars to detect nanohertz gravitational waves from the cosmic background of merging supermassive black holes
• Einstein Telescope / Cosmic Explorer: next-generation ground-based detectors with ten times the sensitivity of LIGO

Key Insights

• Linearized gravity produces a wave equation: gravitational perturbations propagate at the speed of light
• Gravitational waves have two polarizations ($+$ and $\times$), rotated 45 degrees relative to each other, the signature of a spin-2 field
• Only time-varying quadrupole (or higher) moments generate gravitational radiation, there is no gravitational dipole radiation
• The quadrupole formula $h \sim G\ddot{I}/c^4r$ shows that enormous masses in violent motion are needed to produce detectable waves
• Binary pulsars provided the first indirect evidence; LIGO provided the first direct detection in 2015
• Gravitational wave astronomy opens a new window on the universe, probing the most extreme gravitational environments

Looking Ahead

We have seen how general relativity describes the local geometry around isolated masses (Schwarzschild, Kerr) and the radiation of that geometry (gravitational waves). In the final lesson of this phase, we apply Einstein's equations to the universe as a whole. The cosmological solutions: the Friedmann-Lemaitre-Robertson-Walker metrics, describe an expanding universe and set the stage for modern cosmology.