An Introduction to Gravitational Wave Theory

TL;DR

This work surveys the theoretical foundations of gravitational waves from general relativity to their detection and the characterization of CBC waveforms. It develops a perturbative framework (weak-field, post-M Minkowski and post-Newtonian regimes), clarifies gauge issues and the TT decomposition that isolate the two radiative degrees of freedom with helicities $\pm 2$, and derives the quadrupole formula governing GW emission and energy loss. It connects GW generation to sources via multipole expansions and retarded solutions, and explains GW detection through interferometers with detector pattern functions, including the influence of cosmology on amplitude and frequency through redshift and luminosity distance. The notes also discuss GW propagation in curved space-times, the memory effect, and the potential of GW observations to constrain cosmology (standard sirens), highlighting methodological frameworks like the Landau–Lifshitz approach and averaging procedures for gauge-invariant energetics. Collectively, they provide a rigorous, gauge-aware path from GR to observable CBC waveforms and cosmological applications.

Abstract

Introduction to the theoretical foundations of gravitational waves: from general relativity to detection and binary system waveforms. Lecture notes prepared for the MaNiTou summer school on gravitational waves. Draft chapter for the CNRS contemporary Encyclopaedia Sciences to be published by ISTE.

Figures (17)

• Figure 1: Sketch of a binary system of masses $m_{1,2}$ with conserved orbital angular momentum $\vec{J}$ and inclination $\iota$, at a distance $R$ from different detectors (LVK, ET and LISA). The approximate frequency bands of each detector are indicated.
• Figure 2: Qualitative behaviour of the inspiral, merger and ringdown phases of a CBC with their corresponding gravitational waveform as a function of time. There is no analytic method that can reproduce this signal entirely, and different approximation schemes are used. Linearized GR and the PN expansion that will be explained here can be used for the initial inspiral phase, and extrapolated to get a first estimate of the merging amplitude.
• Figure 3: Conservation laws on stationary spacetimes. Evaluating the surface integral \ref{['Komar']} on the innermost surface $S_1$ gives a quantity proportional to the energy-momentum of the planet encompassed. Evaluating it on $S_2$ gives the total energy-momentum of all stars and planets. The difference between the two surface integral is proportional to the energy-momentum of the region between them. Finally since there is no source outside $S_2$, integrating on $S_2$ or $S_3$ gives the same result.
• Figure 4: Rotation between the orbital plane of a binary system and the direction of the observer. The black reference frame is adapted to the orbit, with $\hat{z}$ aligned with the angular momentum, and $\hat{x}$ with the pericenter $P$. The first rotation along $\hat{y}$ by an angle $\theta$ rotates the frame to the blue one, and the second rotation along $\hat{z}$ by an angle $\varphi$ rotates it to the final red frame. The red frame has $\hat{z}$ pointing in the direction of the observer, and $\hat{y}$ axis along the line of nodes, pointing towards the descending node. In the red frame, the longitude of the ascending node $A$ is $\Omega=3\pi/2$, and the longitude of pericenter is $\omega$.
• Figure 5: The effect of the two polarizations on a circular distribution of test masses. In the upper panel (red circles) $h_+\neq 0$ and $h_\times =0$. The lower panel has $h_\times \neq 0$ and $h_+ =0$.