×

zbMATH — the first resource for mathematics

Complete solution of Hadamard’s problem for the scalar wave equation on Petrov type III space-times. (English) Zbl 0951.35131
This very interesting paper is devoted to the solution of Hadamard’s problem on Petrov type III space-times, for the conformally invariant scalar wave equation \[ \square u+ 1/6 Ru= 0,\tag{1} \] and the non-selfadjoint scalar wave equation \[ \square u+ A^a\partial_a u+ Cu= 0, \] where \(\square\) is the Laplace-Beltrami operator corresponding to the metric \(g_{ab}\) of the background space-time \(V_4\), \(u\) is the unknown function, \(R\) is the Ricci scalar, \(A^a\) – the components of a given contravariant vector field and \(C\) – a given scalar function. The background manifold, metric tensor, vector field and scalar function are assumed to be \(C^\infty\). The main result is that there exists no Petrov type III space-times on which the conformally invariant scalar wave equation (1) satisfies Huygens’ principle. Also there exist no Petrov type III space-times on which the non-selfadjoint scalar wave equation satisfies Huygens’ principle.

MSC:
35Q75 PDEs in connection with relativity and gravitational theory
83C20 Classes of solutions; algebraically special solutions, metrics with symmetries for problems in general relativity and gravitational theory
53Z05 Applications of differential geometry to physics
Software:
Maple; NP; NPspinor
PDF BibTeX XML Cite
Full Text: Numdam EuDML arXiv
References:
[1] W.G. Anderson , R.G. McLenaghan and F.D. Sasse , Huygens’ principle for the non-self-adjoint scalar wave equation on Petrov type III space-times , Ann. Inst. Henri Poincaré , Phys. Théor. 70 ( 1999 ) 259 - 276 . Numdam | MR 1718182 | Zbl 0956.83005 · Zbl 0956.83005 · numdam:AIHPA_1999__70_3_259_0 · eudml:76815
[2] L. Asgeirsson , Some hints on Huygens’ principle and Hadamard’s conjecture , Comm. Pure Appl. Math. 9 ( 1956 ) 307 - 326 . MR 82034 | Zbl 0074.31101 · Zbl 0074.31101 · doi:10.1002/cpa.3160090304
[3] J. Carminati and R.G. McLenaghan , Determination of all Petrov type N space-times on which the conformally invariant scalar wave equation satisfies Huygens’ principle , Phys. Lett. 105A ( 1984 ) 351 - 354 . MR 766032
[4] 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 , in: Proceedings of the Journées Relativistes 1984 , Aussois , France , edited by Laboratoire Gravitation et Cosmologie Relativistes, Institut Henri Poincaré , Lecture Notes in Physics , vol. 212 , Springer , Berlin , 1984 . MR 780225 | Zbl 0557.53046 · Zbl 0557.53046
[5] 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. 44 ( 1986 ) 115 - 153 . Numdam | MR 839281 | Zbl 0595.35067 · Zbl 0595.35067 · numdam:AIHPA_1986__44_2_115_0 · eudml:76314
[6] J. Carminati and R.G. McLenaghan , An explicit determination of spacetimes on which the conformally invariant scalar wave equation satisfies Huygens’ principle. Part II: Petrov type D space-times , Ann. Inst. Henri Poincaré , Phys. Théor. 47 ( 1987 ) 337 - 354 . Numdam | MR 933681 | Zbl 0694.35074 · Zbl 0694.35074 · numdam:AIHPA_1987__47_4_337_0 · eudml:76382
[7] J. Carminati and R.G. McLenaghan , An explicit determination of space-times on which the conformally invariant scalar wave equation satisfies Huygens’ principle . Part III: Petrov type III space-times, Ann. Inst. Henri Poincaré , Phys. Théor. 48 ( 1988 ) 77 - 96 . Numdam | MR 947160 | Zbl 0706.35131 · Zbl 0706.35131 · numdam:AIHPA_1988__48_1_77_0 · eudml:76391
[8] J. Carminati , S.R. Czapor , R.G. McLenaghan and G.C. Williams , Consequences of the validity of Huygens’ principle for the conformally invariant scalar wave equation, Weyl’s neutrino equation and Maxwell’s equations on Petrov type II space-times , Ann. Inst. Henri Poincaré , Phys. Théor. 54 ( 1991 ) 9 - 16 . Numdam | MR 1102968 | Zbl 0729.35075 · Zbl 0729.35075 · numdam:AIHPA_1991__54_1_9_0 · eudml:76524
[9] S.R. Czapor , Solving algebraic equations: Combining Buchberger’s algorithm with multivariate factorization , J. Symbolic Comput. 7 ( 1989 ) 49 - 53 . MR 984270 | Zbl 0668.68033 · Zbl 0668.68033 · doi:10.1016/S0747-7171(89)80005-7
[10] S.R. Czapor and R.G. McLenaghan , NP: A Maple package for performing calculations in the Newman-Penrose formalism , Gen. Rel. Gravit. 19 ( 1987 ) 623 - 635 . MR 892637 | Zbl 0613.53033 · Zbl 0613.53033 · doi:10.1007/BF00762558
[11] S.R. Czapor , R.G. McLenaghan and J. Carminati , The automatic conversion of spinor equations to dyad form in MAPLE , Gen. Rel. Gravit. 24 ( 1992 ) 911 - 928 . MR 1180241 | Zbl 0758.53047 · Zbl 0758.53047 · doi:10.1007/BF00759122
[12] J.-C. Faugère , Résolution des systemes d’équation algébriques , Ph.D. Thesis , Université Paris , 1994 .
[13] K.O. Geddes , S.R. Czapor and G. Labahn , Algorithms for Computer Algebra , Kluwer , Norwell, MA , 1992 . MR 1256483 | Zbl 0805.68072 · Zbl 0805.68072
[14] P. Günther , Zur Gültigkeit des huygensschen Prinzips bei partiellen Differentialgleichungen von normalen hyperbolischen Typus , S.-B. Sachs. Akad. Wiss. Leipzig Math.-Natur. K. 100 ( 1952 ) 1 - 43 . MR 50136 | Zbl 0046.32201 · Zbl 0046.32201
[15] J. Hadamard , Lectures on Cauchy’s Problem in Linear Differential Equations , Yale University Press , New Haven , 1923 . JFM 49.0725.04 · JFM 49.0725.04
[16] J. Hadamard , The problem of diffusion of waves , Ann. of Math. 35 ( 1942 ) 510 - 522 . MR 6809 | Zbl 0063.01841 · Zbl 0063.01841 · doi:10.2307/1968806
[17] M. Mathisson , Le probléme de M. Hadamard relatif à la diffusion des ondes , Acta Math. 71 ( 1939 ) 249 - 282 . MR 728 | Zbl 0022.22802 · Zbl 0022.22802 · doi:10.1007/BF02547756
[18] R.G. McLenaghan , An explicit determination of the empty space-times on which the wave equation satisfies Huygens’ principle , Proc. Cambridge Philos. Soc. ( 1969 ). MR 234700 | Zbl 0182.13403 · Zbl 0182.13403
[19] R.G. McLenaghan and F.D. Sasse , Nonexistence of Petrov type III space-times on which Weyl’s neutrino equation or Maxwell’s equations satisfy Huygens’ principle , Ann. Inst. Henri Poincaré , Phys. Théor. 65 ( 1996 ) 253 - 271 . Numdam | MR 1420704 | Zbl 0869.53061 · Zbl 0869.53061 · numdam:AIHPA_1996__65_3_253_0 · eudml:76742
[20] R.G. McLENAGHAN and T.F. Walton , An explicit determination of the non-selfadjoint wave equations on a curved space-time that satisfies Huygens’ principle . Part I: Petrov type N background space-times, Ann. Inst. Henri Poincaré , Phys. Théor. 48 ( 1988 ) 267 - 280 . Numdam | MR 950268 | Zbl 0645.53047 · Zbl 0645.53047 · numdam:AIHPA_1988__48_3_267_0 · eudml:76400
[21] R.G. McLenaghan and T.G.C. Williams , An explicit determination of the Petrov type D space-times on which Weyl’s neutrino equation and Maxwell’s equations satisfy Huygens’ principle , Ann. Inst. Henri Poincaré , Phys. Théor. 53 ( 1990 ) 217 - 223 . Numdam | MR 1079778 | Zbl 0709.53053 · Zbl 0709.53053 · numdam:AIHPA_1990__53_2_217_0 · eudml:76501
[22] M.B. Monagan , K.O. Geddes , K.M. Heal , G. Labahn and S. Vorkoetter , Maple V Programming Guide , Springer , New York , 1996 .
[23] B. Rinke and V. Wünsch , Zum Huygensschen Prinzip bei der skalaren Wellengleichung , Beit. zur Analysis 18 ( 1981 ) 43 - 75 . MR 650138 | Zbl 0501.53010 · Zbl 0501.53010
[24] F.D. Sasse , Huygens’ principle for relativistic wave equations on Petrov type III space-times , Ph.D. Thesis , University of Waterloo , 1997 . · Zbl 0885.53078
[25] T.F. Walton , The validity of Huygens’ principle for the non-self-adjoint scalar wave equations on curved space-time , Master’s Thesis , University of Waterloo , 1988 .
[26] V. Wünsch , Über selbstadjungierte Huygenssche Differentialgleichungen mit vier unabhängigen Variablen , Math. Nachr. 47 ( 1970 ) 131 - 154 . MR 298221 | Zbl 0211.40803 · Zbl 0211.40803 · doi:10.1002/mana.19700470116
[27] V. Wünsch , Huygens’ principle on Petrov type D space-times , Ann. Physik 46 ( 1989 ) 593 - 597 . MR 1051239 | Zbl 0697.53027 · Zbl 0697.53027 · doi:10.1002/andp.19895010806
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.