Spinning Particles in Scalar-Tensor Gravity

1. Chih-Hung Wang National Central University D. A. Burton, R. W.Tucker & C. H. Wang, Physics Letter A 372 (2008) Spinning Particles in Scalar-Tensor Gravity

2. Introduction Equations of motion (EOM) of spinning particles and extended bodies in general relativity have been developed by Papapetrou (1951) and later on by Dixon (1970-1973). It turns out that pole-dipole EOM cannot form a complete system and require extra equations in order to solve them. These extra equations correspond to determine the centre-of-mass world line. Dixon’s multipole analysis has been generalized to Riemann-Cartan space-time by using differential forms, Cartan structure equations, and Fermi-coordinates. (Tucker 2004).

3. We apply this method with given constitutive relations to derive pole-dipole EOM of spinning particles in scalar-tensor gravity with torsion. The solution of pole-dipole EOM in weak field limit is also obtained.

4. Generalized Fermi-normal Coordinates Fermi-normal coordinates are constructed on the open neighbourhood U of a time-like proper-time parametrized curve (). The construction is following: I. Set up orthonormal frames { X} on () satisfying X0 = and use generalized Fermi derivative II. At any point p on , use spacelike autoparallels (): to label all of the points on U of p. III. Parallel-transport orthonormal co-frames { ea } along () from () to U. () v P U

5. Using Cartan structure equations the components of { ea } and connection 1-forms { ab} with respect to Fermi coordinates { } can be expressed in terms of torsion tensor, curvature tensor and their radial derivative evaluated on 

6. In the following investigation, we only need initial values where denotes 4-acceleration of  and are spatial rotations of spacelike orthonormal frames { X1, X2, X3}.

7. Relativistic Balance Laws We start from an action of matter fields in a background spacetime with metric g, metric-compatible connection , and background Brans-Dicke scalar field . The 4-form is constructed tensorially from and, regardless the detailed structure of , it follows The precise details of the sources (stress 3-forms , spin 3-forms and 0-form ) depend on the details of . By imposing equations of motion for and considering has compact support , we obtain

8. Using with straightforward calculation gives Noether identities These equations can be considered as conservation laws of energy-momentum and angular momentum.

9. Equations of motion for a spinning particle To describe the dynamics of a spinning particle, instead of giving details of , we substitute a simple constitutive relations to Noether identities. When we consider a trivial background fields, Minkowski spacetime with equal constant, the model can give a standard result: a spinning particle follows a geodesic carrying a Fermi-Walker spin vector.

10. By constructing Fermi-normal coordinates such that and { e1, e2, e3 } is Fermi-parallel on , Noether identities become where

11. The above system is supplemented by the Tulczyjew-Dixon (subsidiary) conditions We would expected to obtain an analytical solution in arbitrary background fields. We are interested in a spinning particle moving in a special background: Brans-Dicke torsion field with weak-field limit, i.e. neglecting spin-curvature coupling. In this background, we obtain a particular solution and it immediately gives i.e. the spinning particle moving along an autoparallel with parallel-transport of spin vector with respect to along .

12. Conclusion We offer a systematic approach to investigate equations of motion for spinning particles in scalar-tensor gravity with torsion. Fermi-normal coordinates provides some advantages, especially for examining Newtonian limit and simplifying EOM. In background Brans-Dicke torsion field, we obtained spinning particles following autoparallels with parallel-transport of spin vector in weak-field regions. This result has been used to calculate the precession rates of spin vector in weak Kerr-Brans-Dicke spacetime and it leads to the same result (in the leading order) as Lens-Thirring and geodesic precession in weak Kerr space-time (Wang 06).

13. A straightforward generalization is to consider charged spinning particles and include background electromagnetic field.

14. Thank you!