Book contents
- Frontmatter
- Contents
- Foreword
- Acknowledgments
- Introduction
- Notation
- 1 Superluminal motion in the quasar 3C273
- 2 Curved spacetime and SgrA*
- 3 Parallel transport and isometry of tangent bundles
- 4 Maxwell's equations
- 5 Riemannian curvature
- 6 Gravitational radiation
- 7 Cosmological event rates
- 8 Compressible fluid dynamics
- 9 Waves in relativistic magnetohydrodynamics
- 10 Nonaxisymmetric waves in a torus
- 11 Phenomenology of GRB supernovae
- 12 Kerr black holes
- 13 Luminous black holes
- 14 A luminous torus in gravitational radiation
- 15 GRB supernovae from rotating black holes
- 16 Observational opportunities for LIGO and Virgo
- 17 Epilogue: GRB/XRF singlets, doublets? Triplets!
- Appendix A Landau's derivation of a maximal mass
- Appendix B Thermodynamics of luminous black holes
- Appendix C Spin–orbit coupling in the ergotube
- Appendix D Pair creation in a Wald field
- Appendix E Black hole spacetimes in the complex plan
- Appendix F Some units, constants and numbers
- References
- Index
6 - Gravitational radiation
Published online by Cambridge University Press: 17 August 2009
- Frontmatter
- Contents
- Foreword
- Acknowledgments
- Introduction
- Notation
- 1 Superluminal motion in the quasar 3C273
- 2 Curved spacetime and SgrA*
- 3 Parallel transport and isometry of tangent bundles
- 4 Maxwell's equations
- 5 Riemannian curvature
- 6 Gravitational radiation
- 7 Cosmological event rates
- 8 Compressible fluid dynamics
- 9 Waves in relativistic magnetohydrodynamics
- 10 Nonaxisymmetric waves in a torus
- 11 Phenomenology of GRB supernovae
- 12 Kerr black holes
- 13 Luminous black holes
- 14 A luminous torus in gravitational radiation
- 15 GRB supernovae from rotating black holes
- 16 Observational opportunities for LIGO and Virgo
- 17 Epilogue: GRB/XRF singlets, doublets? Triplets!
- Appendix A Landau's derivation of a maximal mass
- Appendix B Thermodynamics of luminous black holes
- Appendix C Spin–orbit coupling in the ergotube
- Appendix D Pair creation in a Wald field
- Appendix E Black hole spacetimes in the complex plan
- Appendix F Some units, constants and numbers
- References
- Index
Summary
“To explain all nature is too difficult a task for any one man or even for any one age.’ Tis much better to do a little with certainty, and leave the rest for others that come after you, than to explain all things.”
Isaac Newton (1642–1727), in G. Simmons, Calculus Gems.Hyperbolic spacetimes possess a local causal structure described by a light cone at every point. The metric obeys the second-order Einstein equations containing one parameter: the velocity of light. This suggests that infinitesimal perturbations of the metric itself propagate along the very same light cones. We have a separation theorem: gravitational radiation propagates in curved spacetime according to a four-covariant wave-equation, in response to which the metric evolves in the tangent bundle. The result is independent of the foliation of spacetime in spacelike hypersurfaces.
Recall that general relativity embodies the Newtonian gravitational potential energy embedded in the metric tensor. Gravitational radiation will be a novel feature which, for finite amplitudes, hereby carries off energy and momentum. As with waves in any field theory, the energy-momentum transport scales with the frequency and amplitude squared.
Gravitational radiation is a spin-2wave, characterized by rotational symmetry over π in the plane orthogonal to the direction of propagation in the spin-classification of M. Fierz and W. Pauli [184]. The lowest-order mass-moment producing gravitational radiation, therefore, is the quadrupole moment. In this chapter, we derive the classical expressions for quadrupole emissions.
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- Publisher: Cambridge University PressPrint publication year: 2005