Generation of Post-Newtonian Gravitational Radiation via Direct Integration of the Relaxed Einstein Equations

Solutions are given representing two independent particles uniformly accelerated in opposite directions; the accelerations are not produced by nodal singularities, which are absent. One solution is constructed from the solution of Bonnor & Swaminarayan ( Z . Phys . 177, 240 (1964)) for two pairs of uniformly accelerated particles by a limiting procedure. Other solutions are obtained by solving the Einstein equations directly. A general solution representing two accelerating particles with arbitrary multipole structure attached to nodal singularities is first given. Then a condition restricting multipole moments is found, and this causes the nodal singularities to disappear. Although solutions of this type do not involve very physical sources, they belong to the best model space-times available today for examining the general theory of the asymptotic structure and the theory of gravitational radiation. Owing to the boost-rotation symmetry, the ADM 4-momentum at spatial infinity vanishes.

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Gravitational waves and the radiative field

10.1093/oso/9780198786399.003.0053 ◽

2018 ◽

Author(s):

Nathalie Deruelle ◽

Jean-Philippe Uzan

Keyword(s):

General Relativity ◽

General Solution ◽

Gravitational Waves ◽

Radiation Field ◽

Gravitational Radiation ◽

Gravitational Force ◽

Energy Momentum Tensor ◽

Momentum Tensor ◽

Einstein Equations ◽

Maxwell Theory

This chapter turns to the gravitational radiation produced by a system of massive objects. The discussion is confined to the linear approximation of general relativity, which is compared with the Maxwell theory of electromagnetism. In the first part of the chapter, the properties of gravitational waves, which are the general solution of the linearized vacuum Einstein equations, are studied. Next, it relates these waves to the energy–momentum tensor of the sources creating them. The chapter then turns to the ‘first quadrupole formula’, giving the gravitational radiation field of these sources when their motion is due to forces other than the gravitational force.

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Gravitational radiation

10.1093/oso/9780198786399.003.0054 ◽

2018 ◽

Author(s):

Nathalie Deruelle ◽

Jean-Philippe Uzan

Keyword(s):

General Relativity ◽

Linear Approximation ◽

Gravitational Radiation ◽

A Priori ◽

Einstein Equations ◽

Maxwell Theory ◽

Radiated Energy ◽

Gravitational Analog ◽

Gravitating Systems ◽

Quadratic Order

This chapter attempts to calculate the radiated energy of a source in the linear approximation of general relativity to infinity in the lowest order. For this, the chapter first expands the Einstein equations to quadratic order in metric perturbations. It reveals that the radiated energy is then given by the (second) quadrupole formula, which is the gravitational analog of the dipole formula in Maxwell theory. This formula is a priori valid only if the motion of the source is due to forces other than gravity. Finally, this chapter shows that, to prove this formula for the case of self-gravitating systems, the Einstein equations to quadratic order must be solved, and the radiative field in the post-linear approximation of general relativity obtained.

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