Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The basics of gravitational wave theory

Eanna E. Flanagan, Scott A. Hughes

TL;DR

This article provides a concise, tutorial-level treatment of gravitational waves in general relativity, starting from linearized theory and progressing to waves in curved backgrounds and their energy content via the Isaacson tensor. It derives the quadrupole formula for GW emission from slow-motion sources, discusses gauge-invariant decompositions to separate radiative from nonradiative degrees of freedom, and explains how detectors respond to GWs through tidal strains and the Riemann tensor. A geometric optics framework is presented to define gravitational waves in curved spacetime and to quantify their energy-momentum, while a broad survey highlights the major high-, low-, very-low-, and ultra-low-frequency GW sources and detectors (e.g., LIGO, LISA). The work emphasizes the potential of GW observations to probe otherwise inaccessible astrophysical processes and early-Universe physics, laying a foundation for interpreting detections and guiding future instrumentation.

Abstract

Einstein's special theory of relativity revolutionized physics by teaching us that space and time are not separate entities, but join as ``spacetime''. His general theory of relativity further taught us that spacetime is not just a stage on which dynamics takes place, but is a participant: The field equation of general relativity connects matter dynamics to the curvature of spacetime. Curvature is responsible for gravity, carrying us beyond the Newtonian conception of gravity that had been in place for the previous two and a half centuries. Much research in gravitation since then has explored and clarified the consequences of this revolution; the notion of dynamical spacetime is now firmly established in the toolkit of modern physics. Indeed, this notion is so well established that we may now contemplate using spacetime as a tool for other science. One aspect of dynamical spacetime -- its radiative character, ``gravitational radiation'' -- will inaugurate entirely new techniques for observing violent astrophysical processes. Over the next one hundred years, much of this subject's excitement will come from learning how to exploit spacetime as a tool for astronomy. This article is intended as a tutorial in the basics of gravitational radiation physics.

The basics of gravitational wave theory

TL;DR

This article provides a concise, tutorial-level treatment of gravitational waves in general relativity, starting from linearized theory and progressing to waves in curved backgrounds and their energy content via the Isaacson tensor. It derives the quadrupole formula for GW emission from slow-motion sources, discusses gauge-invariant decompositions to separate radiative from nonradiative degrees of freedom, and explains how detectors respond to GWs through tidal strains and the Riemann tensor. A geometric optics framework is presented to define gravitational waves in curved spacetime and to quantify their energy-momentum, while a broad survey highlights the major high-, low-, very-low-, and ultra-low-frequency GW sources and detectors (e.g., LIGO, LISA). The work emphasizes the potential of GW observations to probe otherwise inaccessible astrophysical processes and early-Universe physics, laying a foundation for interpreting detections and guiding future instrumentation.

Abstract

Einstein's special theory of relativity revolutionized physics by teaching us that space and time are not separate entities, but join as ``spacetime''. His general theory of relativity further taught us that spacetime is not just a stage on which dynamics takes place, but is a participant: The field equation of general relativity connects matter dynamics to the curvature of spacetime. Curvature is responsible for gravity, carrying us beyond the Newtonian conception of gravity that had been in place for the previous two and a half centuries. Much research in gravitation since then has explored and clarified the consequences of this revolution; the notion of dynamical spacetime is now firmly established in the toolkit of modern physics. Indeed, this notion is so well established that we may now contemplate using spacetime as a tool for other science. One aspect of dynamical spacetime -- its radiative character, ``gravitational radiation'' -- will inaugurate entirely new techniques for observing violent astrophysical processes. Over the next one hundred years, much of this subject's excitement will come from learning how to exploit spacetime as a tool for astronomy. This article is intended as a tutorial in the basics of gravitational radiation physics.

Paper Structure

This paper contains 30 sections, 153 equations, 3 figures.

Table of Contents

  1. Introduction: Spacetime and gravitational waves
  2. Why this article?
  3. The basic basics: Gravitational waves in linearized gravity
  4. Globally vacuum spacetimes: Transverse traceless (TT) gauge
  5. Global spacetimes with matter sources
  6. Local regions of spacetime
  7. Interaction of gravitational waves with a detector
  8. The generation of gravitational waves: Putting in the source
  9. Slow-motion sources in linearized gravity
  10. Extension to sources with non-negligible self gravity
  11. Dimensional analysis
  12. Numerical estimates
  13. Linearized theory of gravitational waves in a curved background
  14. Perturbation theory of curved vacuum spacetimes
  15. General definition of gravitational waves: The geometric optics regime
  16. ...and 15 more sections

Figures (3)

  • Figure 1: Lines of force for a purely $+$ GW (left), and for a purely $\times$ GW (right). Figure kindly provided by Kip Thorne; originally published in Ref. science92.
  • Figure 2: Sensitivity goals of the initial LIGO interferometers, and facility limits on the LIGO sensitivity (taken from Ref. snowmass).
  • Figure 3: Orbital configuration of the LISA antenna.