All torsion-free spherical vacuum solutions of the quadratic Poincar� gauge theory of gravity

Abstract

We find all torsion-free, spherically symmetric, vacuum solutions to the theory of gravity recently proposed by Hehl, Ne'eman, Nitsch, and von der Heyde. There are three classes of solutions: (A) the Schwarzschild metrics with arbitrary mass,M, and arbitrary cosmological constant, Λ; (B) the Nariai-Bertotti metrics with arbitrary positive cosmological constant, Λ; and (C) the conformally flat metrics whose conformai factor is Ω2/ρ 2 where Ω is a function of only the time coordinate τ, and the radial coordinateρ, and satisfies the wave equation in these variables. Hence there is no Birkhoff theorem for this theory. In fact, solutions (C) include some asymptotically flat but nonstationary solutions. On the other hand, solutions (A) include a gravitational confinement potential, as was sought by Hehl et al., since when Λ<0, the weak field limit of the Schwarzschild metric becomes a harmonic oscillator potential. We also discuss the relationship of this theory to the Eddington theory, the Lichnerowicz-Kilmister-Newman-Yang theory, the Nordstrom theory and the Einstein theory with a cosmological constant.