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Cauchy’s problem and Huygens’ principle for relativistic higher spin wave equations in an arbitrary curved space-time. (English) Zbl 0564.35091
Relativistic spin s(s\(\geq 1/2)\), nonzero mass equations are given which in an arbitrary curved space-time are internally consistent. By means of Riesz’ integration method a representation theorem for the solution of Cauchy’s problem, using the constraints of the Cauchy data on the initial hypersurface and suitable ”Green’s formulas,” is proved. Finally, a necessary and sufficient condition for the validity of Huygens’ principle is stated from which it follows that only in space-times of constant curvature do the field equations satisfy Huygens’ principle.

MSC:
35Q99 Partial differential equations of mathematical physics and other areas of application
35C15 Integral representations of solutions to PDEs
83C15 Exact solutions to problems in general relativity and gravitational theory
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[1] Buchdahl, H. A. (1958).Nuovo Cimento,10, 96. · Zbl 0082.22005
[2] Buchdahl, H. A. (1962).Nuovo amento,25, 486. · Zbl 0126.44306
[3] Buchdahl, H. A. (1982).J. Phys. A: Math. Gen.,15, 1, 1057.
[4] Bruhat, Y. (1962). The Cauchy problem, inGravitation; An Introduction to Current Research, L. Witten, ed. (Wiley, New York).
[5] Bruhat, Y. (1964).Ann. Mat. Pura AppL,64, 191. · Zbl 0173.36501
[6] Chevalier, M. (1970).Ann. Inst. Henri Poincaré A,XII(1), 71.
[7] Combet, E. (1965). Solutions elémentaires des dalembertiens généralisées,Mém. Sc. Math. Fasc. CLX (Gauthiers-Villars, Paris). · Zbl 0127.04907
[8] De Witt, B. S., and Brehme, R. W. (1960).Ann. Phys. (N. Y.),9, 220. · Zbl 0092.45003
[9] De Witt, B. S. (1975).Phys. Rep. C,19(6), 295.
[10] Fierz, M., and Pauli, W. (1939).Proc. R. Soc. London Ser. A,173, 211. · Zbl 0023.43004
[11] Friedlander, F. G. (1975).The Wave Equation on a Curved Space-Time (Cambridge University Press, Cambridge). · Zbl 0316.53021
[12] Günther, P. (1965).Wiss. Z. KMU Leipzig,14, 497.
[13] Günther, P. (1975).Math. Nachr.,69, 39. · Zbl 0335.35064
[14] Günther, P. (1975).ZAMM,55, 205. · Zbl 0324.53012
[15] Günther, P., and Wünsch, V. (1974).Math. Nachr.,63, 97. · Zbl 0288.35042
[16] Hadamard, J. (1923).Lectures on Cauchy’s Problem in Linear Partial Differential Equations (Yale University Press, New Haven). · JFM 49.0725.04
[17] Infeld, I., and van der Waerden, B. L. (1933). S.Ber. Preuss. Akad. Wiss. Berlin, 380.
[18] Lax, P. D., and Phillips, R. S. (1978).Commun. Pure Appl. Math.,31, 415. · Zbl 0378.35039
[19] Lichnerowicz, A. (1964).Bull. Soc. Math. France,92, 11.
[20] Mc Lenaghan, R. G. (1974).Ann. Inst. Henri Poincaré A,20, 153.
[21] Penrose, R. (1960).Ann. Phys. (N.Y.),10, 171. · Zbl 0091.21404
[22] Penrose, R. (1965).Proc. R. Soc. London Ser. A,284, 159. · Zbl 0129.41202
[23] Pirani, F. A. E. (1965). InLectures on General Relativity, Brandeis Summer Institute in Theoretical Physics (Prentice-Hall, Englewood Cliffs, New Jersey). · Zbl 0176.55402
[24] Riesz, M. (1949).Acta Math.,81, 1. · Zbl 0033.27601
[25] Roman, P. (1961).Theory of Elementary Particles (North-Holland Publ. Company, Amsterdam). · Zbl 0103.44403
[26] Schimming, R. (1978).Beiträge zur Analysis,11, 45.
[27] Schmutzer, E. (1968).Relativistische Physik (Teubner-Verlag, Leipzig). · Zbl 0193.57301
[28] Schouten, J. A., and Struik, D. J. (1935). Einführung in die neueren Methoden der Differentialgeometric I, II. Groningen. · Zbl 0011.17404
[29] Treder, H. J. (1973).Ann. Phys. (Leipzig),30, 229.
[30] Wünsch, V. (1976).C. R. Acad. Sci. Paris, Sér. A.,283, 983.
[31] Wünsch, V. (1978,1979).Beiträge zur Analysis,12, 47, 13, 147.
[32] Wünsch, V. (1980).Math. Nachr.,94, 211. · Zbl 0439.35040
[33] Wünsch, V. (1979).Math. Nachr.,89, 321. · Zbl 0407.53017
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