Global exterior Cauchy problem for \(\text{spin }3/2\) zero rest-mass fields in the Schwarzschild space-time. (English) Zbl 0878.35115

Nicolas, Jean-Philippe

Commun. Partial Differ. Equations 22, No. 3-4, 465-502 (1997).

The author sets out to study spin 3/2 fields from the point of view of hyperbolic partial differential equations. The purpose of the paper is to set up a technical basis which will allow further analytic investigations in the future. He chooses a particular Ricci-flat space-time, the Schwarzschild black hole, on which he solves the global Cauchy problem for the Dirac equation, \[ \nabla^{AA'} \sigma^C_{A'B'} =0, \] for the first potential of a spin 3/2 zero rest-mass field, for solutions with minimum regularity. This study is a first step towards the understanding of more difficult questions like the development of a time-dependent scattering theory for spin 3/2 fields on black hole space-times.

Reviewer: A.D.Osborne (Keele)

MSC:

35Q75	PDEs in connection with relativity and gravitational theory
83C05	Einstein’s equations (general structure, canonical formalism, Cauchy problems)
83C57	Black holes
83C60	Spinor and twistor methods in general relativity and gravitational theory; Newman-Penrose formalism

Keywords:

spin 3/2 fields; Schwarzschild black hole; global Cauchy problem; Dirac equation

References:

[1]	Bachelot A., Ann. Inst. Henri Poincaré 54 pp 261– (1991)
[2]	DOI: 10.1007/BF02859608 · Zbl 0082.22005
[3]	Chandrasekhar S., Oxford Science Publications Publications, Oxford University Press, 1983.
[4]	DOI: 10.1098/rspa.1936.0111 · Zbl 0014.42304
[5]	DOI: 10.1098/rspa.1939.0140 · Zbl 0023.43004
[6]	Gelrsquo;Fand I.M., Amer. Math. Soc. Transl. 2 pp 207– (1956)
[7]	Hille E., American Mathematical Society colloquium Publications 31 pp 349– (1965)
[8]	Mason L.J., Twistor Newsletter 37 pp 1– (1994)
[9]	Nicolas J.-P., Ann. Inst Henri Poincaré 62 pp 145– (1995)
[10]	Penrose, R. ”Twistors as spin 3/2 charges Gravitation and modern cosmology”. Edited by: Zichichi, A., de Sabbata, N. and Sánchez, N. 129–137.
[11]	DOI: 10.1098/rspa.1965.0058 · Zbl 0129.41202
[12]	Penrose R, Spinors and space-time Cambridge mongraphs on mathematical physics, Two-spinor calculus and relativistic fields 1 (1984)
[13]	Penrose R, Spinors and space-time, Cambridge monographs on mathematical physics, Spinor and twistor methods in spacetime geometry 2 (1986)
[14]	DOI: 10.1103/PhysRev.60.61 · Zbl 0026.28704