On detecting the gravitomagnetic field of the earth by means of orbiting clocks

Based on the recent finding that the difference in proper time of two clocks in prograde and retrograde equatorial orbits about the Earth is of the order 10^{-7}s per revolution, the possibility of detecting the terrestrial gravitomagnetic field by means of clocks carried by satellites is discussed. A mission taking advantage of this influence of the rotating Earth on the proper time is outlined and the conceptual difficulties are briefly examined.Comment: Talk given at the 32nd COSPAR Scientific Assembly, held at Nagoya, Japan, 12-19 July 1998, 4 pages LATE

An alternative derivation of the gravitomagnetic clock effect

The possibility of detecting the gravitomagnetic clock effect using artificial Earth satellites provides the incentive to develop a more intuitive approach to its derivation. We first consider two test electric charges moving on the same circular orbit but in opposite directions in orthogonal electric and magnetic fields and show that the particles take different times in describing a full orbit. The expression for the time difference is completely analogous to that of the general relativistic gravitomagnetic clock effect in the weak-field and slow-motion approximation. The latter is obtained by considering the gravitomagnetic force as a small classical non-central perturbation of the main central Newtonian monopole force. A general expression for the clock effect is given for a spherical orbit with an arbitrary inclination angle. This formula differs from the result of the general relativistic calculations by terms of order c^{-4}.Comment: LaTex2e, 11 pages, 1 figure, IOP macros. Submitted to Classical and Quantum Gravit

On the Possibility of Measuring the Gravitomagnetic Clock Effect in an Earth Space-Based Experiment

In this paper the effect of the post-Newtonian gravitomagnetic force on the mean longitudes $l$ of a pair of counter-rotating Earth artificial satellites following almost identical circular equatorial orbits is investigated. The possibility of measuring it is examined. The observable is the difference of the times required to $l$ in passing from 0 to 2$\pi$ for both senses of motion. Such gravitomagnetic time shift, which is independent of the orbital parameters of the satellites, amounts to 5$\times 10^{-7}$ s for Earth; it is cumulative and should be measured after a sufficiently high number of revolutions. The major limiting factors are the unavoidable imperfect cancellation of the Keplerian periods, which yields a constraint of 10$^{-2}$ cm in knowing the difference between the semimajor axes $a$ of the satellites, and the difference $I$ of the inclinations $i$ of the orbital planes which, for $i\sim 0.01^\circ$, should be less than $0.006^\circ$. A pair of spacecrafts endowed with a sophisticated intersatellite tracking apparatus and drag-free control down to 10$^{-9}$ cm s$^{-2}$ Hz$^{-{1/2}}$ level might allow to meet the stringent requirements posed by such a mission.Comment: LaTex2e, 22 pages, no tables, 1 figure, 38 references. Final version accepted for publication in Classical and Quantum Gravit

Spinning test particles and clock effect in Schwarzschild spacetime

We study the behaviour of spinning test particles in the Schwarzschild spacetime. Using Mathisson-Papapetrou equations of motion we confine our attention to spatially circular orbits and search for observable effects which could eventually discriminate among the standard supplementary conditions namely the Corinaldesi-Papapetrou, Pirani and Tulczyjew. We find that if the world line chosen for the multipole reduction and whose unit tangent we denote as $U$ is a circular orbit then also the generalized momentum $P$ of the spinning test particle is tangent to a circular orbit even though $P$ and $U$ are not parallel four-vectors. These orbits are shown to exist because the spin induced tidal forces provide the required acceleration no matter what supplementary condition we select. Of course, in the limit of a small spin the particle's orbit is close of being a circular geodesic and the (small) deviation of the angular velocities from the geodesic values can be of an arbitrary sign, corresponding to the possible spin-up and spin-down alignment to the z-axis. When two spinning particles orbit around a gravitating source in opposite directions, they make one loop with respect to a given static observer with different arrival times. This difference is termed clock effect. We find that a nonzero gravitomagnetic clock effect appears for oppositely orbiting both spin-up or spin-down particles even in the Schwarzschild spacetime. This allows us to establish a formal analogy with the case of (spin-less) geodesics on the equatorial plane of the Kerr spacetime. This result can be verified experimentally.Comment: IOP macros, eps figures n. 2, to appear on Classical and Quantum gravity, 200