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

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

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!

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!

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, 061101(R) (2007). The generalization to arbitrary-spin particles: O.V. Teryaev, arXiv:hep-ph/9904376 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).

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)].

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

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)].

Mathisson-Papapetrou equations or Myron Mathisson

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)

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

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!

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

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

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!

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.

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

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

Pomeransky-Khriplovich approach can be verified for a rotating frame

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).

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!

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

Comparison of classical and quantum gravitational spin effects A.J. Silenko and O.V. Teryaev, Phys. Rev. D 71, 064016 (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:

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

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:

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

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

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.

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).

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, 061101(R) (2007). The equation of spin motion coincides with the Gorbatsevich-Mashhoon equation:

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

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

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

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, 064016 (2005). Motion of momentum and spin differs in a static gravitational field and a uniformly accelerated frame but the helicity evolution coincides!

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

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 :

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)

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/9904376

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)

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!

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

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

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