Spinning Particles in Scalar-Tensor Gravity

Chih-Hung Wang

National Central University

D. A. Burton, R. W.Tucker & C. H. Wang, Physics Letter A 372 (2008)

Introduction

Equations of motion (EOM) of spinning particles and Equations of motion (EOM) of spinning particles and extended bodies in general relativity have been developed extended bodies in general relativity have been developed by Papapetrou (1951) and later on by Dixon (1970-1973). by Papapetrou (1951) and later on by Dixon (1970-1973). It turns out that pole-dipole EOM cannot form a complete It turns out that pole-dipole EOM cannot form a complete system and require extra equations in order to solve them. system and require extra equations in order to solve them. These extra equations correspond to determine the These extra equations correspond to determine the centre-of-mass world line.centre-of-mass world line.

Dixon’s multipole analysis has been generalized to Dixon’s multipole analysis has been generalized to Riemann-Cartan space-time by using differential forms, Riemann-Cartan space-time by using differential forms, Cartan structure equations, and Fermi-coordinates. Cartan structure equations, and Fermi-coordinates. (Tucker 2004).(Tucker 2004).

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

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.

P

v

()

U

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

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 }.

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

Using

with straightforward calculation gives Noether identities

These equations can be considered as conservation laws of energy-momentum and angular momentum.

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.

By constructing Fermi-normal coordinates such that and

{ e1, e2, e3 } is Fermi-parallel on , Noether identities become

where

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 .

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).

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