×

zbMATH — the first resource for mathematics

On purely radiative space-times. (English) Zbl 0584.53038
The existence of space-times representing pure gravitational radiation which comes in from infinity and interacts with itself is discussed. They are characterized as solutions of Einstein’s vacuum field equations possessing a smooth structure at past null infinity which forms the ”future null cone at past timelike infinity with complete generators.” The ”pure radiation problem” is analysed where ”free initial data” for Einstein’s field equations are prescribed on the null cone at past time- like infinity. It is demonstrated how the pure radiation problem can be formulated as a local initial value problem for the symmetric hyperbolic system of reduced conformal vacuum field equations. Its solutions are uniquely determined by the free data.

MSC:
53C80 Applications of global differential geometry to the sciences
83C30 Asymptotic procedures (radiation, news functions, \(\mathcal{H} \)-spaces, etc.) in general relativity and gravitational theory
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Bondi, H., van der Burg, M. G. J., Metzner, A. W. K.: Gravitational waves in general relativity. VII Waves from axi-symmetric isolated systems. Proc. Roy. Soc. Lond. A269, 21-52 (1962) · Zbl 0106.41903 · doi:10.1098/rspa.1962.0161
[2] Bruhat, Y.: Problème des conditions initiales sur un conoide charactéristique. C.R. Acad. Sci. 3971-3973 (1963) · Zbl 0118.23006
[3] Friedlander, F. G.: The wave-equation on a curved space-time. Cambridge: Cambridge University Press 1975 · Zbl 0316.53021
[4] Friedrich, H.: Eine Untersuchung der Einsteinschen Vakuumfeldgleichungen in der Umgebung regulärer und singulärer Nullhyperflächen. Thesis, University of Hamburg (1979)
[5] Friedrich, H.: The asymptotic characteristic initial value problem for Einstein’s vacuum field equations as an initial value problem for a first order quasilinear symmetric hyperbolic system. Proc. Roy. Soc. Lond. A378, 401-421 (1981) · Zbl 0481.58026 · doi:10.1098/rspa.1981.0159
[6] Friedrich, H.: Cauchy problems for the conformal vacuum field equations in general relativity. Commun. Math. Phys.91, 445-472 (1983) · Zbl 0555.35116 · doi:10.1007/BF01206015
[7] Friedrich, H.: On the hyperbolicity of Einstein’s and other gauge field equations. Commun. Math. Phys.100, 525-543 (1985) · Zbl 0588.35058 · doi:10.1007/BF01217728
[8] Friedrich, H.: The pure radiation problem for analytic data. (In preparation)
[9] Friedrich, H., Stewart, J.: Characteristic initial data and wave front singularities in general relativity. Proc. Roy. Soc. Lond. A385, 345-371 (1983) · Zbl 0513.58043 · doi:10.1098/rspa.1983.0018
[10] Geroch, R.: Asymptotic Structure of Space-Time. In: Esposito, F. R., Written, L. (eds.). Asymptotic structure of space-time. New York: Plenum Press 1976
[11] Geroch, R., Horowitz, G. T.: Asymptotically simple does not imply asymptotically minkowskian. Phys. Rev. Lett.40, 203-206 (1978) · doi:10.1103/PhysRevLett.40.203
[12] Newman, E., Penrose, R.: An approach to gravitational radiation by a method of spin coefficients. J. Math. Phys.3, 566-578 (1962) · Zbl 0108.40905 · doi:10.1063/1.1724257
[13] Newman, E., Penrose, R.: Note on the Bondi-Metzner-Sachs group. J. Math. Phys.7, 863-870 (1966) · doi:10.1063/1.1931221
[14] Penrose, R.: Null hypersurface initial data for classical fields of arbitrary spin and for general relativity, published 1963 in Aerospace Res. Lab. Tech. Doc. Rep. 63-56 (P. G. Bergmann) reprinted in G.R.G.12, 225-264 (1980)
[15] Penrose, R.: Asymptotic properties of fields and space-times. Phys. Rev. Lett.10, 66-68 (1963) · doi:10.1103/PhysRevLett.10.66
[16] Penrose, R.: Conformal treatment of infinity. In: DeWitt, DeWitt (eds.). Relativity, groups and topology. New York: Gordon & Breach 1964 · Zbl 0148.46403
[17] Penrose, R.: Zero rest-mass fields including gravitation: Asymptotic behaviour. Proc. Roy. Soc. Lond. A284, 159-203 (1965) · Zbl 0129.41202 · doi:10.1098/rspa.1965.0058
[18] Penrose, R.: Structure of space-time. In: DeWitt, C. M., Wheeler, J. A. (eds.). Battelle rencontres. New York: W. A. Benjamin 1967
[19] Penrose, R.: Relativistic symmetry groups. In: Barut, A. O. (ed.). Group theory in non-linear problems. Amsterdam: D. Reidel 1974 · Zbl 0281.22021
[20] Pirani, F. A. E.: Invariant formulation of gravitational radiation theory. Phys. Rev.105, 1089-1099 (1957) · Zbl 0077.41901 · doi:10.1103/PhysRev.105.1089
[21] Sachs, R. K.: Gravitational waves in general relativity. VI The outgoing radiation condition. Proc. Roy. Soc. Lond A264, 309-338 (1961) · Zbl 0098.19204 · doi:10.1098/rspa.1961.0202
[22] Sachs, R. K.: Gravitational waves in general relativity. VIII Waves in asymptotically flat space-time. Proc. Roy. Soc. Lond. A270, 103-126 (1962) · Zbl 0101.43605 · doi:10.1098/rspa.1962.0206
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.