Book contents
- Frontmatter
- Contents
- List of exercises
- List of projects
- Preface
- How to use this book
- 1 Special relativity
- 2 Scalar and electromagnetic fields in special relativity
- 3 Gravity and spacetime geometry: the inescapable connection
- 4 Metric tensor, geodesics and covariant derivative
- 5 Curvature of spacetime
- 6 Einstein's field equations and gravitational dynamics
- 7 Spherically symmetric geometry
- 8 Black holes
- 9 Gravitational waves
- 10 Relativistic cosmology
- 11 Differential forms and exterior calculus
- 12 Hamiltonian structure of general relativity
- 13 Evolution of cosmological perturbations
- 14 Quantum field theory in curved spacetime
- 15 Gravity in higher and lower dimensions
- 16 Gravity as an emergent phenomenon
- Notes
- Index
9 - Gravitational waves
Published online by Cambridge University Press: 05 June 2012
- Frontmatter
- Contents
- List of exercises
- List of projects
- Preface
- How to use this book
- 1 Special relativity
- 2 Scalar and electromagnetic fields in special relativity
- 3 Gravity and spacetime geometry: the inescapable connection
- 4 Metric tensor, geodesics and covariant derivative
- 5 Curvature of spacetime
- 6 Einstein's field equations and gravitational dynamics
- 7 Spherically symmetric geometry
- 8 Black holes
- 9 Gravitational waves
- 10 Relativistic cosmology
- 11 Differential forms and exterior calculus
- 12 Hamiltonian structure of general relativity
- 13 Evolution of cosmological perturbations
- 14 Quantum field theory in curved spacetime
- 15 Gravity in higher and lower dimensions
- 16 Gravity as an emergent phenomenon
- Notes
- Index
Summary
Introduction
One of the key new phenomena that arises in general relativity is the existence of solutions to Einstein's equations which represent disturbances in the spacetime that propagate at the speed of light. Such solutions are called gravitational waves and this chapter will explore several features of them.
Propagating modes of gravity
Within the context of special relativity, it is easy to identify a wave solution. For example, a propagating, monochromatic spherical wave will be described by an amplitude that varies in space and time as f(t, r) ∞ r−1 exp[−iω(t − r)]. This disturbance clearly propagates from the origin with the speed of light (which is unity in our notation) with an amplitude that decreases as (1/r). Since the energy flux of a wave varies as the square of the amplitude, this wave transports a constant amount of energy across every spherical surface. Such a description can be easily made Lorentz covariant in terms of an appropriate wave vector, etc., and has an unambiguous meaning.
The situation is somewhat more complicated in the case of gravity for two (closely related) reasons. First, not all the components of the metric gab enjoy equal status in the dynamics of gravity. We saw in Section 6.3 that the g00 and g0α components do not propagate in general relativity. The equations governing them are constraint equations involving and and are analogous to the equation governing the gauge dependent mode in electrodynamics.
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- Chapter
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- GravitationFoundations and Frontiers, pp. 399 - 451Publisher: Cambridge University PressPrint publication year: 2010