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Spinning Particles in Scalar-Tensor Gravity

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					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
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).
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.
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
                                                             ()
                                                             v

II. At any point p on , use spacelike autoparallels             P

    ():

     to label all of the points on U of p.
                                                                    U
III. Parallel-transport orthonormal co-frames { ea } along
() from () to 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.

				
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