Spinning Particles in Scalar-Tensor Gravity

1
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)

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.
• Dixons 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!