An explicit determination of the space-times on which the conformally invariant scalar wave equation satisfies Huygens’ principle. III: Petrov type III space-times. (English) Zbl 0706.35131

Carminati, J.; McLenaghan, R. G.

Ann. Inst. Henri Poincaré, Phys. Théor. 48, No. 1, 77-96 (1988).

Summary: [For part I, see ibid. 44, 115-153 (1986; Zbl 0595.35067).]
It is shown that the validity of Huygens’ principle for the conformally invariant scalar wave equation, Maxwell’s equations or Weyl’s neutrino equation on a Petrov type III space-time, implies that the space-time is conformally related to one on which the repeated principal null vector field of the Weyl tensor, is recurrent. Further, it is proven tat if a certain mild assumption which is shown to be suggested by the necessary conditions, is imposed on the covariant derivative of the Weyl tensor, then there are no Petrov type III space-times on which any of the above equations satisfies Huygens’ principle.

MSC:

35Q75	PDEs in connection with relativity and gravitational theory
35L05	Wave equation
83C99	General relativity

Keywords:

Huygens’ principle; conformally invariant scalar wave equation; Maxwell’s equations; Weyl’s neutrino equation

Citations:

Zbl 0595.35067

Software:

NP; Maple

References:

[1]	B. Buchberger , A Survey on the Method of Gröbner Bases for Solving Problems in Connection with Systems of Multi-variate Polynomials . Article in The Second RIKEN International Symposium on Symbolic and Algebraic Computation by Computers, edited by N. Inada and T. Soma. World Scientific Publishing Co ., Singapore , 1984 , p. 69 - 83 . MR 835951 | Zbl 0645.68057 · Zbl 0645.68057
[2]	J. Carminati and R.G. Mclenaghan , Some new results on the validity of Huygens’ principle for the scalar wave equation on a curved space-time . Article in Gravitation, Geometry and Relativistic Physics, Proceedings of the Journées Relativistes 1984 , Aussois , France, edited by Laboratoire Gravitation et Cosmologie Relativistes. Institut Henri Poincaré ;. Lectures Notes in Physics , t. 212 , Springer-Verlag , Berlin , 1984 . MR 780225 | Zbl 0557.53046 · Zbl 0557.53046
[3]	J. Carminati and R.G. Mclenaghan , Determination of all Petrov type N spacetimes on which the conformally invariant scalar wave equation satisfies Huygens’ principle . Phys. Lett. , t. 105 A, 1984 , p. 351 - 354 . MR 766032 · Zbl 0694.35074
[4]	J. Carminati and R.G. Mclenaghan , An explicit determination of the Petrov type N space-times on which the conformally invariant scalar wave equation satisfies Huygens’ principle . Ann. Inst. Henri Poincaré, Phys. Théor. , t. 44 , 1986 , p. 115 - 153 . Numdam | MR 839281 | Zbl 0595.35067 · Zbl 0595.35067
[5]	J. Carminati and R.G. Mclenaghan , The validity of Huygens’ principle for the conformally invariant scalar wave equation, Maxwell’s equations and Weyl’s neutrino equation on Petrov type D and type III space-times . Phys. Lett. , t. 118 A, 1986 , p. 322 - 324 . MR 867005
[6]	J. Carminati and R.G. Mclenaghan , An explicit determination of the space-times on which the conformally invariant scalar wave equation satisfies Huygens’ principle. Part II: Petrov type D space-times . Ann. Inst. Henri Poincaré, Phys. Théor. , in press. Numdam | Zbl 0694.35074 · Zbl 0694.35074
[7]	B.W. Char , K.O. Geddes , W.M. Gentleman , G.H. Gonnet , The design of MAPLE: a compact, portable and powerful computer algebra system , Proc. EU ROCAL , 83 . Lecture Notes in Computer Science , t. 162 , 1983 , p. 101 .
[8]	S.R. Czapor and R.G. Mclenaghan , NP: A MAPLE package for performing calculations in the Newman-Pensose formalism . Gen. Rel. Grav. , t. 19 , 1987 , p. 623 - 635 . MR 892637 | Zbl 0613.53033 · Zbl 0613.53033
[9]	R. Debever , Le rayonnement gravitationnel : le tenseur de Riemann en relativité générale . Cah. Phys. , t. 168 - 169 , 1964 , p. 303 - 349 . MR 187877
[10]	P. Günther , Zur Gültigkeit des Huygensschen Princips bei partiellen Differentialgleichungen von normalen hyperbolischen Typus . S.-B. Sachs. Akad. Wiss. Leipzig Math.-Natur. K. , t. 100 , 1952 , p. 1 - 43 . MR 50136 | Zbl 0046.32201 · Zbl 0046.32201
[11]	P. Gunther , Ein Beispiel einer nichttrivalen Huygensschen Differentialgleichungen mit vier unabhängigen Variablen . Arch. Rational Mech. Anal. , t. 18 , 1965 , p. 103 - 106 . MR 174865 | Zbl 0125.05404 · Zbl 0125.05404
[12]	P. Gunther , Eigine Satze uber Huygenssche Differentialgleichungen . Wiss. Zeitschr. Karl Marx Univ., Math.-Natu. Reihe Leipzig , t. 14 , 1965 , p. 498 - 507 . MR 198012 | Zbl 0173.12203 · Zbl 0173.12203
[13]	P. Günther and V. Wünsch , Maxwellsche Gleichungen und Huygenssches Prinzip I . Math. Nach. , t. 63 , 1974 , p. 97 - 121 . MR 363377 | Zbl 0288.35042 · Zbl 0288.35042
[14]	J. Hadamard , Lectures on Cauchy’s problem in linear partial differential equations . Yale University Press , New Haven , 1923 . JFM 49.0725.04 · JFM 49.0725.04
[15]	H.P. Künzle , Maxwell fields satisfying Huygens’ principle . Proc. Cambridge Philos. Soc. , t. 64 , 1968 , p. 770 - 785 .
[16]	R.G. Mclenaghan , An explicit determination of the empty space-times on which the wave equation satisfies Huygens’ principle . Proc. Cambridge Philos. Soc. , t. 65 , 1969 , p. 139 - 155 . MR 234700 | Zbl 0182.13403 · Zbl 0182.13403
[17]	R.G. Mclenaghan and J. Leroy , Complex recurrent space-times . Proc. Roy. Soc. London , t. A 327 , 1972 , p. 229 - 249 . MR 309517 | Zbl 0243.53030 · Zbl 0243.53030
[18]	R.G. Mclenaghan , On the validity of Huygens’ principle for second order partial differential equations with four independent variables. Part I: Derivation of necessary conditions . Ann. Inst. Henri Poincaré , t. A 20 , 1974 , p. 153 - 188 . Numdam | MR 361452 | Zbl 0287.35058 · Zbl 0287.35058
[19]	R.G. Mclenaghan , Huygens’ principle . Ann. Inst. Henri Poincaré , t. A 27 , 1982 , p. 211 - 236 . Numdam | MR 694586 | Zbl 0528.35057 · Zbl 0528.35057
[20]	E.T. Newman and R. Penrose , An approach to gravitational radiation by a method of spin coefficients . J. Math. Phys. , t. 3 , 1962 , p. 566 - 578 . MR 141500 | Zbl 0108.40905 · Zbl 0108.40905
[21]	R. Penrose , A spinor approach to general relativity . Ann. Physics , t. 10 , 1960 , p. 171 - 201 . MR 115765 | Zbl 0091.21404 · Zbl 0091.21404
[22]	A.Z. Petrov , Einstein-Raume . Academie Verlag , Berlin , 1964 . MR 162594
[23]	F.A.E. Pirani , Introduction to gravitational radiation theory. Article in Lectures on General Relativity , edited by S. Deser and W. Ford, Brandeis Summer Institute in Theoretical Physics , t. 1 , 1964 , Prentice-Hall , New York .
[24]	B. Rinke and V. Wünsch , Zum Huygensschen Prinzip bei der skalaren Wellengleichung . Beitr. zur Analysis , t. 18 , 1981 , p. 43 - 75 . MR 650138 | Zbl 0501.53010 · Zbl 0501.53010
[25]	V. Wünsch , Über selbstadjungierte Huygenssche Differentialgleichungen mit vier unabhängigen Variablen . Math. Nach. , t. 47 , 1970 , p. 131 - 154 . MR 298221 | Zbl 0211.40803 · Zbl 0211.40803
[26]	V. Wünsch , Maxwellsche Gleichungen und Huygenssches Prinzip II . Math. Nach. , t. 73 , 1976 , p. 19 - 36 . MR 426807 | Zbl 0288.35043 · Zbl 0288.35043
[27]	V. Wünsch , Über eine Klasse Konforminvarianter Tensoren . Math. Nach. , t. 73 , 1976 , p. 37 - 58 . MR 433342 | Zbl 0287.53014 · Zbl 0287.53014
[28]	V. Wünsch , Cauchy-problem und Huygenssches Prinzip bei einigen Klassen spinorieller Feldgleichungen I . Beitr. zur Analysis , t. 12 , 1978 , p. 47 - 76 . MR 507097 | Zbl 0448.58022 · Zbl 0448.58022
[29]	V. Wünsch , Cauchy-problem und Huyenssches Prinzip bei einigen Klassen spinorieller Feldgleichungen II . Beitr. zur Analysis , t. 13 , 1979 , p. 147 - 177 . MR 536225 | Zbl 0467.35067 · Zbl 0467.35067