We develop a first-order approximation method for the influence of spin on the motion of extended spinning test masses in a gravitational field. This approach is illustrated for approximately circular equatorial motion in the exterior Kerr spacetime. In this case, the analytic results for the first-order approximation are compared to the numerical integration of the exact system and the limitations of the first-order results are pointed out. Furthermore, we employ our analytic results to illustrate the gravitomagnetic clock effect for spinning particles.

This comment is in response to Nieves and Pal [1] who dispute our claim [2] that a classical gravitational field may possibly distinguish between Dirac and Majorana neutrinos, described in terms of Gaussian wave packets propagating in a Lense-Thirring background, where the distinction is manifested in spin-gravity corrections to the neutrino oscillation length. They contend that our model for Majorana neutrinos is incorrect and that any conclusions relating to this neutrino type are not reliable. Furthermore, they suggest that any distinction between the two neutrino types will be suppressed by factors of m/E where m is the neutrino mass and E is its mean energy of propagation, and claim that this distinction is unobservable when m/E ≪ 1. We beg to disagree.As noted above, our model for the Dirac and Majorana neutrinos is motivated by a wave packet approach in quantum mechanics, as opposed to a plane-wave expansion in quantum field theory. This is an essential detail which Nieves and Pal fail to acknowledge. In addition, the gravitational field [2] is incorporated in terms of a gravitational phase Φ G , giving rise to an interaction Hamiltonian H ΦG with spin-dependent features, to be evaluated in terms of time-independent perturbation theory. These two details are important for framing the context underpinning our reply. Regarding the technical concerns, we agree that the Majorana condition they note in their eq. (1) is certainly true for a fermion field operator. However, we again emphasize that our perspective is quantum mechanical, so our treatment of the Majorana condition must be described in terms of wave functions. Adopting their notation, our approach is to identify [3,4] Unlike what Nieves and Pal claim, it indeed follows [3] that W 1,2 is a solution of the free particle equation / k W 1,2 = ± m W 1,2 [5] for wave functions. However, a more precise implementation of (1) with our notation leads to a Majorana wave packet model with the formwhereand ξ(k) is the Gaussian function [2]. Clearly, it follows from (2) that |ψ c 1(2) Maj. = ± |ψ 1(2) Maj. . This leads to a modification of our eq. (13) for the Majorana matrix element [2], which isOur plots [2], as applied to the SN 1987A data, are completely unaffected by the adjustments because the corrections only alter the contributions coupled to M ΩR 2 /r 2 , which are all exponentially damped compared to the M/r contributions. As for the m/E suppression issue raised by Nieves and Pal, this is of no relevance because all such terms are automatically excluded within the construction of |W 1,2 (k) Maj. , so everything presented in [2] is of leading order. As shown in (5), and also present in our eq. (13)

It is shown that the Pauli-Lubanski spin vector defined in terms of curvilinear co-ordinates does not satisfy Lorentz invariance for spin-1/2 particles in noninertial motion along a curved trajectory. The possibility of detecting this violation in muon decay experiments is explored, where the noninertial contribution to the decay rate becomes large for muon beams with large momenta and trajectories with radius of curvature approaching the muon's Compton wavelength scale. A new spacelike spin vector is derived from the Pauli-Lubanski vector that satisfies Lorentz invariance for both inertial and noninertial motion. In addition, this spin vector suggests a generalization for the classification of spin-1/2 particles, and has interesting properties that are applicable for both massive and massless particles.

We study the relative motion of nearby free test particles in cosmological spacetimes, such as the FLRW and LTB models. In particular, the influence of spatial inhomogeneities on local tidal accelerations is investigated by means of simple spherically symmetric models. The implications of our results for the dynamics of the solar system are briefly discussed. That is, on the basis of the models studied in this paper, we estimate the tidal influence of the cosmic gravitational field on the orbit of the Earth around the Sun and show that the corresponding temporal rate of variation of the astronomical unit is negligibly small.

Dwarf galaxies and globular clusters may contain intermediate mass black holes (10 3 to 10 5 M ⊙ ) in their cores. Estimates of ∼ 10 3 neutron stars in the central parsec of the Galaxy and similar numbers in small elliptical galaxies and globular clusters along with an estimated high probability of ms-pulsar formation in those environments has led many workers to propose the use of ms-pulsar timing to measure the mass and spin of intermediate mass black holes. Models of pulsar motion around a rotating black hole generally assume geodesic motion of a "test" particle in the Kerr metric. These approaches account for well-known effects like de Sitter precession and the Lense-Thirring effect but they do not account for the non-linear effect of the pulsar's stress-energy tensor on the space-time metric. Here we model the motion of a pulsar near a black hole with the Mathisson-Papapetrou-Dixon (MPD) equations. Numerical integration of the MPD equations for black holes of mass 2 × 10 6 , 10 5 and 10 3 M ⊙ shows that the pulsar will not remain in an orbital plane with motion vertical to the plane being largest relative to the orbit's radial dimensions for the lower mass black holes. The pulsar's out of plane motion will lead to timing variations that are up to ∼ 10µs different from those predicted by planar orbit models. Such variations might be detectable in long term observations of millisecond pulsars. If pulsar signals are used to measure the mass and spin of intermediate mass black holes on the basis of dynamical models of the received pulsar signal then the out of plane motion of the pulsar should be part of that model.