Solutions Stationnaires en Th\'eorie de Kaluza-Klein

Abstract

Kaluza-Klein theory is a 5-dimensional Einstein general relativity; it has the interest of describing on an equal footing the laws of gravitation and electromagnetism in a geometrically unified way. We present it in Chapter 1, and we generalize it by adding to the equations of the theory the Lanczos tensor (endowed with the same physical properties as the Einstein tensor but quadratic with respect to the Riemann tensor.) One has obtained in the last decade all the spherically symmetric 2-stationary solutions (independent of time and of the extra coordinate) of the ``special" Kaluza-Klein theory. The study of the stability of these solutions against radial excitations is carried out in Chapter 2. We begin by presenting the spherically symmetric 2-static solutions; then we write and separate the perturbation-linearized equations of the 5-metric. The problem of stability against small oscillations is reduced to an eigenvalue problem which we discuss in detail in the static-solution parameter space. We show that regular solutions of non-vanishing finite energy (Kaluza-Klein solitons) --with non-euclidean spatial topology-- are stable. A broad class of singular solutions, containing among others the Schwarzschild solution, are also stable. Finally our stability results are compared to those obtained previously by Tomimatsu for a less broad class of solutions. We search in Chapter 3 for other exact stationary solutions, endowed this time with cylindrical symmetry, thus actually 4-stationary (depending on only one spacelike coordinate). First we obtain by a systematic study all the 4-stationary solutions of the special theory, some of which are interpreted as neutral or charged distributional cosmic string sources. We generalize these solutions in a following section by considering the Lanczos tensor, and we find