# Particle in Gravitational Field of a Rotating Body

#### Abstract

Effective Lagrangian describing gravitational source spin-particle spin interactions is given. Cosmological and astrophysical consequences of such interaction are examined. Although stronger than expected, the spin-spin interactions do not change any cosmological effect observed so far. They are important for background primordial neutrinos. TUM-HEP-209/94 UPR-644-T/94 MPI-PTh-95/34 December 1994 Introduction It is not uncommon to find in the Universe rotating massive objects. This rotation may be rather slow, like in the case of Earth, or relatively rapid, like that of some neutron stars. Spinning of the source changes the resulting gravitational field and introduces new with respect to the case of simple static sources, angular momentum dependent, gravitational forces. We find it interesting to study and clarify the status of the source-spin dependent gravitational interactions between source and particles travelling through its field. We consider a simplest case of nonzero-spin particle, a spin-1/2 fermion. To be specific, in this note we shall discuss in some detail neutrino interacting with the spinning Sun, but in fact our effective Lagrangian introduced in Section 2. is a general one, valid for any kind of spin-half fermion and any rotating source, like a pulsar or a rotating black hole. 1.Gravitational field of a rotating body The gravitational field of a spinning sphere of mass M and angular momentum ~ J = M~a is described by the Kerr metric, which is an exact solution to the Einstein equations. Since we are going to apply methods of Minkowski space field theory to interactions of spin 1 2 fermions with the spinning background, it is meaningful and sufficient to consider the asymptotic form of the Kerr metric obtained in the limit 1 r → 0. As we are going to consider the effects of rotation, we give the asymptotic form of the Kerr metric up to terms ( rg r )2 and ( rg r a r ), where rg = 2M/M 2 P lanck is the Schwarzchild radius, (we use units and notation of [1]) g00 = 1− rg r + ( rg 2r ) gij = −δij(1 + rg r + 1 2 ( rg r )) g0j = rg r3 ωj ωj = (~a× ~x)j (1) The reference frame has been fixed to the axis of rotation of the body and the metric is written in the so-called isotropic coordinates (the coordinate system in which the asymptotic Schwarzchild metric assumes the diagonal form). One should note that the exact Kerr metric gives the upper limit on the radius a, a < rg/2 (cf. [1]). The asymtotic metric (1) can also be obtained, without any reference to the Kerr metric, just by solving the linearized (in weak field approximation) Einstein equations [1]. Our

### Cite this paper

@inproceedings{Lalak1995ParticleIG, title={Particle in Gravitational Field of a Rotating Body}, author={Zygmunt Lalak and Stefan Pokorski and Julius Wess}, year={1995} }