Gravitational waves in general relativity VIII. Waves in asymptotically flat space-time
1962/10/30 by R. K. : Sachs · 1,864 citations
Earth and Planetary Sciences · Mathematics · Physics and Astronomy · #Bounded function #Classical mechanics #Cosmology and Gravitation Theories #General relativity #Geodesic #Geophysics and Gravity Measurements #Gravitation #Gravitational field #Gravitational wave #Mathematical analysis #Mathematics #Metric tensor #Physics #Pulsars and Gravitational Waves Research #Quantum mechanics #Riemann hypothesis
paper · doi:10.1098/rspa.1962.0206
published in Proceedings of the Royal Society of London A Mathematical and Physical Sciences 270(1340), 103-126 (Royal Society)
openalex publication_date 1962/10/30 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/04
Abstract
Abstract Gravitational fields containing bounded sources and gravitational radiation are examined by analyzing their properties at spatial infinity. A convenient way of splitting the metric tensor and the Einstein field equations, applicable in any space-time, is first introduced. Then suitable boundary conditions are set. The group of co-ordinate transformations that preserves the boundary conditions is analyzed. Different possible gravitational fields are characterized intrinsically by a combination of (i) characteristic initial data, and (ii) Dirichlet data at spatial infinity. To determine a particular solution one must specify four functions of three variables and three functions of two variables; these functions are not subject to constraints. A method for integrating the field equations is given; the asymptotic behaviour of the metric and Riemann tensors for large spatial distances is analyzed in detail; the dynamical variables of the radiation modes are exhibited; and a superposition principle for the radiation modes of the gravitational field is suggested. Among the results are: (i) the group of allowed co-ordinate transformations contains the inhomogeneous orthochronous Lorentz group as a subgroup; (ii) each of the five leading terms in an asymptotic expansion of the Riemann tensor has the algebraic structure previously predicted from analyzing the Petrov classification; (iii) gravitational waves appear to carry mass away from the interior; (iv) time-dependent periodic solutions of the field equations which obey the stated boundary conditions do not exist. It was found that the general fields studied in the present work are in many ways very similar to the axially symmetric fields recently studied by Bondi, van der Burg & Metzner.
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