In General Relativity, the rotation of a gravitating body like the Earth influences the motion of orbiting test particles or satellites in a non-Newtonian way. This causes, e.g., a precession of the orbital plane known as the Lense-Thirring effect and a precession of the spin of a gyroscope known as the Schiff effect. Here, we discuss a third effect first introduced by Cohen and Mashhoon called the gravitomagnetic clock effect. It describes the difference in proper time of counterrevolving clocks after a revolution of $2π$. For two clocks on counterrotating equatorial circular orbits around the Earth, the effect is about $10^{-7}$ seconds per revolution, which is quite large. We introduce a general relativistic definition of the gravitomagnetic clock effect which is valid for arbitrary pairs of orbits. This includes rotations in the same direction and different initial conditions, which are crucial if the effect can be detected with existing satellites or with payloads on nondedicated missions. We also derive the post-Newtonian expansion of the general relativistic expression and calculate the effect for the example of a satellite of a Global Navigation Satellite System compared to a geostationary satellite.
The difference in the proper azimuthal periods of revolution of two standard clocks in direct and retrograde orbits about a central rotating mass is proportional to J/Mc^2, where J and M are, respectively, the proper angular momentum and mass of the source. In connection with this gravitomagnetic clock effect, we explore the possibility of using spaceborne standard clocks for detecting the gravitomagnetic field of the Earth. It is shown that this approach to the measurement of the gravitomagnetic field is, in a certain sense, theoretically equivalent to the Gravity Probe-B concept.
Gravitomagnetic clock effects for circularly rotating orbits in black hole spacetimes are studied from a relative observer point of view, clarifying the roles played by special observer families.
The essence of the gravitomagnetic clock effect is properly defined showing that its origin is in the topology of world lines with closed space projections. It is shown that, in weak field approximation and for a spherically symmetric central body, the loss of synchrony between two clocks counter-rotating along a circular geodesic is proportional to the angular momentum of the source of the gravitational field. Numerical estimates are presented for objects within the solar system. The less unfavorable situation is found around Jupiter.
General relativity predicts that two freely counter-revolving test particles in the exterior field of a central rotating mass take different periods of time to complete the same full orbit; this time difference leads to the gravitomagnetic clock effect. The effect has been derived for circular equatorial orbits; moreover, it has been extended via azimuthal closure to spherical orbits around a slowly rotating mass. In this work, a general formula is derived for the main gravitomagnetic clock effect in the case of slow motion along an arbitrary {\it elliptical} orbit in the exterior field of a s…
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.
As a consequence of gravitomagnetism, which is a fundamental weak-field prediction of general relativity and ubiquitous in gravitational phenomena, clocks show a difference in their proper periods when moving along identical orbits in opposite directions about a spinning mass. This time shift is induced by the rotation of the source and may be used to verify the existence of the terrestrial gravitomagnetic field by means of orbiting clocks. A possible mission scenario is outlined with emphasis given to some of the major difficulties which inevitably arise in connection with such a venture.
We have developed a general geometric treatment of the GCE valid for any stationary axisymmetric metric. The method is based on the remark that the world lines of objects rotating along spacely circular trajectories are in any case, for those kind of metrics, helices drawn on the flat bidimensional surface of a cylinder. Applying the obtained formulas to the special cases of the Kerr and weak field metric for a spinning body, known results for time delays and synchrony defects are recovered.
In this article, we discuss how to carryover gravitomagnetic clock effect from classical general relativity to quantum theory and how to calculate this effect in quantum mechanics. Our calculation is valid for semi-classical regime and can be considered as the first step towards a complete gravitomagnetic quantum theory. We also show the analogy between energy levels that corresponds to the clock effect. In fact, it is argued that in quantum mechanics clock effect arises as energy level splitting in gravitomagnetic field.
We study the gravitomagnetic effect in the context of absolute parallelism with the use of a modified geodesic equation via a free parameter b. We calculate the time difference in two atomic clocks orbiting the Earth in opposite directions and find a small correction due to the coupling between the torsion of the spacetime and the internal structure of atomic clocks measured by the free parameter.
We outline a mission with the aim of directly detecting the gravitomagnetic field of the Earth. This mission is called Gravity Probe C. Gravity Probe C(lock) is based on a recently discovered and surprisingly large gravitomagnetic clock effect. The main idea is to compare the proper time of two standard clocks in direct and retrograde orbits around the Earth. After one orbit the proper time difference of two such clocks is predicted to be of the order of $2\times 10^{-7}$ s. The conceptual difficulty to perform Gravity Probe C is expected to be comparable to that of the Gravity Probe B--missio…
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 …
The paper summarizes the most important effects in Einsteinian gravitomagnetic fields related to propagating light rays, moving clocks and atoms, orbiting objects, and precessing spins. Emphasis is put onto the gravitational interaction of spinning objects. The gravitomagnetic field lines of a rotating or spinning object are given in analytic form.
The general relativistic gravitomagnetic clock effect involves a coupling between the orbital motion of a test particle and the rotation of the central mass and results in a difference in the proper periods of two counter-revolving satellites. It is shown that at O(c^-2) this effect has a simple analogue in the electromagnetic case. Moreover, in view of a possible measurement of the clock effect in the gravitational field of the Earth, we investigate the influence of some classical perturbing forces of the terrestrial space environment on the orbital motion of test bodies along opposite trajec…
Using gyroscopes we generalize results, obtained for the gravitomagnetic clock effect in the particular case when the exterior spacetime is produced by a rotating dust cylinder, to the case when the vacuum spacetime is described by the general cylindrically symmetric Lewis spacetime. Results are contrasted with those obtained for the Kerr spacetime.
The main theoretical aspects of gravitomagnetism are reviewed. It is shown that the gravitomagnetic precession of a gyroscope is intimately connected with the special temporal structure around a rotating mass that is revealed by the gravitomagnetic clock effect. This remarkable effect, which involves the difference in the proper periods of a standard clock in prograde and retrograde circular geodesic orbits around a rotating mass, is discussed in detail. The implications of this effect for the notion of ``inertial dragging'' in the general theory of relativity are presented. The theory of the …
In order to detect the gravitomagnetic clock effect by means of two counter-orbiting satellites placed on identical equatorial and circular orbits around the Earth with radius 7000 km their radial and azimuthal positions must be known with an accuracy of delta r =10^{-1} mm and delta phi =10^{-2} mas per revolution. In this work we investigate if the radial and azimuthal perturbations induced by the dynamical and static parts of the Earth' s gravitational field meet this requirements. While the radial direction is affected only by harmonic perturbations with periods up to some tens of days, th…
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$π$ 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 o…
This paper contains a review of the theory and practice of gravitomagnetism, with particular attention to the different and numerous proposals which have been put forward to experimentally or observationally verify its effects. The basics of the gravitoelectromagnetic form of the Einstein equations is expounded. Then the Lense-Thirring and clock effects are described, reviewing the essentials of the theory. Space based and Earth based experiments are listed. Other effects, such as the coupling of gravitomagnetism with spin, are described and orders of magnitude are considered to give an idea o…
The general relativistic gravitomagnetic clock effect consists in the fact that two point particles orbiting a central spinning object along identical, circular equatorial geodesic paths, but in opposite directions, exhibit a time difference in describing a full revolution. It turns out that the particle rotating in the same sense of the central body is slower than the particle rotating in the opposite sense. In this paper it is proposed to measure such effect in an Earth laboratory experiment involving interferometry of slow neutrons. With a sphere of 2.5 cm radius and spinning at 4.3 x 10^4 …
The gravitomagnetic clock effect and the Sagnac effect for circularly rotating orbits in stationary axisymmetric spacetimes are studied from a relative observer point of view, clarifying their relationships and the roles played by special observer families. In particular Semerák's recent characterization of extremely accelerated observers in terms of the two-clock clock effect is shown to be complemented by a similarly special property of the single-clock clock effect.
The general relativistic gravitomagnetic clock effect consists in the fact that two massive test bodies orbiting a central spinning mass in its equatorial plane along two identical circular trajectories, but in opposite directions, take different times in describing a full revolution with respect to an asymptotically inertial observer. In the field of the Earth such time shift amounts to 10^{-7} s. Detecting it by means of a space based mission with artificial satellites is a very demanding task because there are severe constraints on the precision with which the radial and azimuthal positions…
We discuss gravitomagnetism in connection with rotating cylindrical systems. In particular, the gravitomagnetic clock effect is investigated for the exterior vacuum field of an infinite rotating cylinder. The dependence of the clock effect on the Weyl parameters of the stationary Lewis metric is determined. We illustrate our results by means of the van Stockum spacetime.
We discuss the influence of the cosmological constant on the gravitomagnetic clock effect and the gravitational time delay of electromagnetic rays. Moreover, we consider the relative motion of a binary system to linear order in the cosmological constant $Λ$. The general expression for the effect of $Λ$ on pericenter precession is given for arbitrary orbital eccentricity.
The detection of some tiny gravitomagnetic effects in the field of the Earth by means of artificial satellites is a very demanding task because of the various other perturbing forces of gravitational and non-gravitational origin acting upon them. Among the gravitational perturbations a relevant role is played by the Earth solid and ocean tides. In this communication I outline their effects on the detection of the Lense-Thirring drag of the orbits of LAGEOS and LAGEOS II, currently analyzed, and the proposed GP-C experiment devoted to the measurement of the clock effect.
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 …
We study the motion of spinning test particles in Kerr spacetime using the Mathisson-Papapetrou equations; we impose different supplementary conditions among the well known Corinaldesi-Papapetrou, Pirani and Tulczyjew's and analyze their physical implications in order to decide which is the most natural to use. We find that if the particle's center of mass world line, namely the one chosen for the multipole reduction, is a spatially circular orbit (sustained by the tidal forces due to the spin) then the generalized momentum $P$ of the test particle is also tangent to a spatially circular orbit…
After an introduction on phenomena due to spin and mass-energy currents on clocks and photons, we review the 1995-2001 measurements of the gravitomagnetic field of Earth and Lense-Thirring effect obtained by analyzing the orbits of the two laser-ranged satellites LAGEOS and LAGEOS II; this method has provided a direct measurement of Earth's gravitomagnetism with accuracy of the order of 20 %. A future accurate measurement of the Lense-Thirring effect, at the level of 1 % accuracy, may include the LARES experiment that will also provide other basic tests of general relativity and gravitation. F…
Based on a generalized uncertainty principle, Salecker-Wigner inequalities are modified. When applied to black holes, they give a modified black hole lifetime: $T_{\rm MB}\sim\frac{M^3}{m^3_{\rm p}}(1-m^2_{\rm p}/M^2)t_{\rm p}$, and the number of bits required to specify the information content of the black hole as the event horizon area in Planck units: $N\sim\frac{M^2}{m^2_{\rm p}}(1-m^2_{\rm p}/M^2)$.
In this work some aspects of the detection of certain general relativistic effects in the weak gravitational field of the Earth via laser-ranged data to some existing or proposed geodetic satellites are examined. The focus is on the Lense-Thirring drag of the orbit of a test body, the gravitomagnetic clock effect and the gravitoelectric perigee shift. The impact of some sources of systematic errors is investigated. An experiment whose goal is the measurement of the PPN parameters beta and gamma in the terrestrial field with LAGEOS satellites at a level of 10^(-3)is presented. A modified versio…
Quantum interference of "clocks", namely of particles with time-evolving internal degrees of freedom (DOFs), is a promising avenue to test genuine general relativistic effects in quantum systems. The clock acquires which path information while experiencing different proper times on traversing the arms of the interferometer, leading to a drop in its path visibility. We consider scenarios where the clock is subject to environmental noise as it transits through the interferometer. In particular, we develop a generalized formulation of interferometric visibility affected by noise on the clock. We …
In this paper we discuss on recent attempts aimed at demonstrating that, contrary to well-known results, the second clock effect (SCE) does not take place in generalized Weyl spaces -- spaces with arbitrary nonmetricity -- denoted here as $W_4$ spaces. These attempts include Weyl gauge theories of gravity, as well as the symmetric teleparallel theories (STTs). Our approach to this issue is based on the adoption of Weyl gauge symmetry (WGS) which is a manifest symmetry of the basic laws of Weyl geometry. We shall consistently adapt mathematical and geometrical quantities and concepts so that th…
This is the first of a couple of papers in which, by exploiting the capabilities of the Hamiltonian approach to general relativity, we get a number of technical achievements that are instrumental both for a disclosure of \emph{new} results concerning specific issues, and for new insights about \emph{old} foundational problems of the theory. The first paper includes: 1) a critical analysis of the various concepts of symmetry related to the Einstein-Hilbert Lagrangian viewpoint on the one hand, and to the Hamiltonian viewpoint, on the other. This analysis leads, in particular, to a re-interpreta…
The role of non-inertial frames in a class of models of general relativity is clarified by means of Dirac's theory of constraints. The identification of a York canonical basis allows to give the interpretation of the gauge variables as generalized inertial effects and to identify the Dirac observables of the gravitational field with generalized tidal effects. York time is the gauge variable controlling the clock synchronization convention. Differently from special relativity, the instantaneous 3-spaces are dynamically determined.
The gravitational collapse of a star is an important issue both for general relativity and astrophysics, which is related to the well known "frozen star" paradox. This paradox has been discussed intensively and seems to have been solved in the comoving-like coordinates. However, to a real astrophysical observer within a finite time, this problem should be discussed in the point of view of the distant rest-observer, which is the main purpose of this paper. Following the seminal work of Oppenheimer and Snyder (1939), we present the exact solution for one or two dust shells collapsing towards a p…
Aharonov-Kaufherr model of quantum space-time which accounts Reference Frames (RF) quantum effects is considered in Relativistic Quantum Mechanics framework. For RF connected with some macroscopic object its free quantum motion - wave packet smearing results in additional uncertainty of test particle coordinate. Due to the same effects the use of Galilean or Lorentz transformations for this RFs becomes incorrect and the special quantum space-time transformations are introduced. In particular for any RF the proper time becomes the operator in other RF. This time operator calculated solving rela…
