1. Classical and Quantum Spins in Curved Spacetimes
Alexander J. Silenko
Belarusian State University
Myron Mathisson: his life, work, and influence on current research
Warsaw 2007

2. OUTLINE
General properties of spin interactions with gravitational fields
Classical equations of spin motion in curved spacetimes
Comparison between classical and quantum gravitational spin effects
Equivalence Principle and spin

3. General properties of spin interactions with gravitational fields
Anomalous gravitomagnetic moment is equal to zero
Gravitoelectric dipole moment is equal to zero
Spin dynamics is caused only by spacetime metric!

4. Classical and quantum theories are in the best compliance!
Kobzarev – Okun relations
I.Yu. Kobzarev, L.B. Okun, Gravitational Interaction of Fermions.
Zh. Eksp. Teor. Fiz. 43, 1904 (1962) [Sov. Phys. JETP 16, 1343 (1963)].
These relations define form factors at zero momentum transfer
gravitational and inertial masses are equal
anomalous gravitomagnetic moment
is equal to zero
gravitoelectric dipole moment is equal to zero
Classical and quantum theories are in the best compliance!

5. The generalization to arbitrary-spin particles:
The absence of the anomalous gravitomagnetic moment is experimentally checked in:
B. J. Venema, P. K. Majumder, S. K. Lamoreaux, B. R. Heckel, and E. N. Fortson, Phys. Rev. Lett. 68, 135 (1992).
see the discussion in: A.J. Silenko and O.V. Teryaev, Phys. Rev. D 76, (R) (2007).
The generalization to arbitrary-spin particles:
O.V. Teryaev, arXiv:hep-ph/
The absence of the gravitoelectric dipole moment results in the absence of spin-gravity coupling:
see the discussion in: B. Mashhoon, Lect. Notes Phys. 702, 112 (2006).

6. The Equivalence Principle manifests in the general equations of motion of classical particles
and their spins:
A.A. Pomeransky and I.B. Khriplovich, Zh. Eksp. Teor. Fiz. 113, 1537 (1998) [J. Exp. Theor. Phys. 86, 839 (1998)].

7. Classical equations of spin motion in curved spacetimes
Two possible methods of obtaining classical equations of spin motion:
i) search for appropriate covariant equations
Thomas-Bargmann-Mishel-Telegdi equation – linear in spin, electromagnetic field
Good-Nyborg equation – quadratic in spin, electromagnetic field
Mathisson-Papapetrou equations – all orders in spin, gravitational field
ii) derivation of equations with the use of some physical principles
Pomeransky-Khriplovich equations – linear and quadratic in spin, electromagnetic and gravitational fields

8. Good-Nyborg equation is wrong!
The derivation based on the initial Proca-Corben-Schwinger equations for spin-1 particles confirms the Pomeransky-Khriplovich equations
A.J. Silenko, Zs. Eksp. Teor. Fiz. 123, 883 (2003) [J. Exp. Theor. Phys. 96, 775 (2003)].

9. Mathisson-Papapetrou equationsor
Myron Mathisson

10. C. Chicone, B. Mashhoon, and B. Punsly, Phys. Lett. A 343, 1 (2005)
Connection between four-momentum and four-velocity:
Additional force is of second order in the spin
C. Chicone, B. Mashhoon, and B. Punsly, Phys. Lett. A 343, 1 (2005)

11. Pole-dipole approximation
The spin dynamics given by
the Pomeransky-Khriplovich approach
is the same!

12. The momentum dynamics given by the Pomeransky-Khriplovich approach results from the spin dynamics
S is 3-component spin
t is world time
H is Hamiltonian defining the momentum and spin dynamics
The momentum dynamics can be deduced!

13. Pomeransky-Khriplovich approach
Tetrad equations of momentum and spin motion
are Ricci rotation coefficients
Similar to equations of momentum and spin motion of
Dirac particle (g=2) in electromagnetic field
is electromagnetic field tensor

14. Pomeransky-Khriplovich approach
Tetrad variables are blue, t ≡ x0

15. Pomeransky-Khriplovich approach
Pomeransky-Khriplovich approach needs to be grounded
The 3-component spin vector is defined in a particle rest frame. What particle rest frame should be used?
When the metric is nonstatic, covariant and tetrad velocities are equal to zero
(u=0 and u=0)
in different frames!

16. Pomeransky-Khriplovich approach
Local flat Lorentz frame is a natural choice of particle rest frame.
Only the definition of the 3-component spin vector in a flat tetrad frame is consistent with the quantum theory.
Definition of 3-component spin vector in the classical and quantum theories agrees with the Pomeransky-Khriplovich approach
are the Dirac matrices but
are not.

17. Pomeransky-Khriplovich approach
Pomeransky-Khriplovich gravitomagnetic field is nonzero
even for a static metric!

18. Pomeransky-Khriplovich approach
In the reference
A.A. Pomeransky and I.B. Khriplovich, Zh. Eksp. Teor. Fiz. 113,
1537 (1998) [J. Exp. Theor. Phys. 86, 839 (1998)]
the following weak-field approximation was used:
This approximation is right for static metric but
incorrect for nonstatic metric!
Pomeransky-Khriplovich equations agree with quantum theory
resulting from the Dirac equation

19. Pomeransky-Khriplovich approach
can be verified for a rotating frame

20. Gorbatsevich, Exp. Tech. Phys. 27, 529 (1979);
Pomeransky-Khriplovich approach results in the Gorbatsevich-Mashhoon equation
Gorbatsevich, Exp. Tech. Phys. 27, 529 (1979);
Mashhoon, Phys. Rev. Lett. 61, 2639 (1988).
A. J. Silenko (unpublished).

21. Pomeransky-Khriplovich approach
Another exact solution was obtained for a Schwarzschild metric
A. A. Pomeransky, R. A. Senkov, and I. B. Khriplovich, Usp. Fiz. Nauk 43, 1129 (2000) [Phys. Usp. 43, 1055 (2000)].
However, Pomeransky-Khriplovich and Mathisson-Papapetrou
equations of particle motion does not agree with each other!

22. Comparison of classical and quantum gravitational spin effects
Classical and quantum effects should be similar due to the correspondence principle
Niels Bohr

23. Comparison of classical and quantum gravitational spin effects
A.J. Silenko and O.V. Teryaev, Phys. Rev. D 71, (2005).
Silenko and Teryaev establish full agreement between quantum theory based on the Dirac equation and the classical theory
The exact transformation of the Dirac equation for the metric
to the Hamilton form was carried out by Obukhov:

24. Comparison of classical and quantum gravitational spin effects
Yu. N. Obukhov, Phys. Rev. Lett. 86, 192 (2001);
Fortsch. Phys. 50, 711 (2002).
This Hamiltonian covers the cases of a weak Schwarzschild field and a uniformly accelerated frame

25. Comparison of classical and quantum gravitational spin effects
Silenko and Teryaev used the Foldy-Wouthuysen transformation for relativistic particles in external fields and derived the relativistic Foldy-Wouthuysen Hamiltonian:

26. Comparison of classical and quantum gravitational spin effects
Quantum mechanical equations of momentum and spin motion

27. Comparison of classical and quantum gravitational spin effects
Semiclassical equations of momentum and spin motion
Pomeransky-Khriplovich equations give the same result!

28. Comparison of classical and quantum gravitational spin effects
These formulae agree with the results obtained for some particular cases with classical and quantum approaches:
A. P. Lightman, W. H. Press, R. H. Price, and S. A. Teukolsky, Problem book in relativity and gravitation (Princeton Univ. Press, Princeton, 1975).
F. W. Hehl and W. T. Ni, Phys. Rev. D 42, 2045 (1990).
These formulae perfectly describe a deflection of massive and massless particles by the Schwarzschild field.

29. Comparison of classical and quantum gravitational spin effects
Spinning particle in a rotating frame
The exact Dirac Hamiltonian was obtained by Hehl and Ni:
F. W. Hehl and W. T. Ni, Phys. Rev. D 42, 2045 (1990).

30. Comparison of classical and quantum gravitational spin effects
The result of the exact Foldy-Wouthuysen transformation is given by
A.J. Silenko and O.V. Teryaev, Phys. Rev. D 76, (R) (2007).
The equation of spin motion coincides with the Gorbatsevich-Mashhoon equation:

31. Comparison of classical and quantum gravitational spin effects
The particle motion is characterized by the operators of velocity and acceleration:
For the particle in the rotating frame
w is the sum of the Coriolis and centrifugal accelerations

32. Comparison of classical and quantum gravitational spin effects
The classical and quantum approaches are in the best agreement

33. Equivalence Principle and spin
Gravity is geometrodynamics!
The Einstein Equivalence Principle predicts the equivalence of gravitational and inertial effects and states that the result of a local non-gravitational experiment in an inertial frame of reference is independent of the velocity or location of the experiment
Albert Einstein

34. Equivalence Principle and spin
The absence of the anomalous gravitomagnetic and gravitoelectric dipole moments is a manifestation of the Equivalence Principle
Another manifestation of the Equivalence Principle was shown in Ref.
A.J. Silenko and O.V. Teryaev, Phys. Rev. D 71, (2005).
Motion of momentum and spin differs in a static gravitational field and a uniformly accelerated frame but the helicity evolution coincides!

35. Equivalence Principle and spin
φ depends only on but f is a function of both and

36. Equivalence Principle and spin
Dynamics of unit momentum vector n=p/p:
Difference of angular velocities of rotation of
spin and momentum depends only on :

37. Equivalence Principle and spin
Pomeransky-Khriplovich equations assert the exact validity of this statement in strong static gravitational and inertial fields
The unit vectors of momentum and velocity rotate with the same mean frequency in strong static gravitational and inertial fields but instantaneous angular velocities of their rotation can differ
A.J. Silenko and O.V. Teryaev (unpublished)

38. Equivalence Principle and spin
Gravitomagnetic field
Equivalence Principle predicts the following properties:
Gravitomagnetic field making the velocity rotate twice faster than the spin changes the helicity
Newertheless, the helicity of a scattered massive particle is not influenced by the rotation of an astrophysical object
O.V. Teryaev, arXiv:hep-ph/

39. Equivalence Principle and spin
Gravitomagnetic field
Analysis of Pomeransky-Khriplovich equations gives the same results:
Gravitomagnetic field making the velocity rotate twice faster than the spin changes the helicity
Newertheless, the tetrad momentum and the spin rotate with the same angular velocity
Directions of the tetrad momentum and the velocity coincide at infinity
As a result, the helicity of a scattered massive particle is not influenced by the rotation of an astrophysical object
A.J. Silenko and O.V. Teryaev (unpublished)

40. Equivalence Principle and spin
Gravitomagnetic field
Alternative conclusions about the helicity evolution made in several other works
Y.Q. Cai, G. Papini, Phys. Rev. Lett. 66, 1259 (1991)
D. Singh, N. Mobed, G. Papini, J. Phys. A 3, 8329 (2004)
D. Singh, N. Mobed, G. Papini, Phys. Lett. A 351, 373 (2006)
are not correct!

41. Summary Spin dynamics is defined by the Equivalence Principle
Mathisson-Papapetrou and Pomeransky-Khriplovich equations predict the same spin dynamics
Anomalous gravitomagnetic and gravitoelectric dipole moments of classical and quantum particles are equal to zero
Pomeransky-Khriplovich equations define gravitoelectric and gravitomagnetic fields dependent on the particle four-momentum
Behavior of classical and quantum spins in curved spacetimes is the same and any quantum effects cannot appear

42. Summary The classical and quantum approaches are in the best agreement
The helicity evolution in gravitational fields and corresponding accelerated frames coincides, being the manifestation of the Equivalence Principle
Massless particles passing throughout gravitational fields of astrophysical objects does not change the helicity
The evolution of helicity of massive particles passing throughout gravitational fields of astrophysical objects is not affected by their rotation
The classical and quantum approaches are in the best agreement

43. Thank you for attention