We discuss the quantum and classical dynamics of a particle with spin in the gravitational field of a rotating source. A relativistic equation describing the motion of classical spin in curved spacetimes is obtained. We demonstrate that the precession of the classical spin is in a perfect agreement with the motion of the quantum spin derived from the Foldy-Wouthuysen approach for the Dirac particle in a curved spacetime. We show that the precession effect depends crucially on the choice of a tetrad. The results obtained are compared to the earlier computations for different tetrad gauges.
Recently, it was shown that the extreme Kerr black hole is the only candidate for a (Kerr) black hole limit of stationary and axisymmetric, uniformly rotating perfect fluid bodies with a zero temperature equation of state. In this paper, necessary and sufficient conditions for reaching the black hole limit are presented.
A relatively recent study by Mars and Senovilla provided us with a uniqueness result for the exterior vacuum gravitational field of global models describing finite isolated rotating bodies in equilibrium in General Relativity (GR). The generalisation to exterior electrovacuum gravitational fields, to include charged rotating objects, is presented here.
We study different models of radiating slowly rotating bodies up to the first order in the angular velocity. It is shown that up to this order the evolution of the eccentricity is highly model-dependent even for very compact objects.
We show that a primeval seed magnetic field arises due to spin-degeneracy breaking of fermions caused by the dragging of inertial frames in the curved spacetime of rotating astrophysical bodies. This seed magnetic field would arise even due to electrically neutral fermions such as neutrons. As an example, we show that an ideal neutron star rotating at $500$ revolutions per second, having mass $0.83$ M$_{\odot}$ and described by an ensemble of degenerate neutrons, would have $0.12$ Gauss seed magnetic field at its center arising through the breaking of spin-degeneracy.
How do test bodies move in scalar-tensor theories of gravitation? We provide an answer to this question on the basis of a unified multipolar scheme. In particular, we give the explicit equations of motion for pointlike, as well as spinning test bodies, thus extending the well-known general relativistic results of Mathisson, Papapetrou, and Dixon to scalar-tensor theories of gravity. We demonstrate the validity of the equivalence principle for test bodies.
We study the exterior vacuum problem for first and second order stationary and axially symmetric perturbations of static bodies. The boundary conditions and their compatibility for the existence of an asymptotically flat exterior solution are discussed.
The gravitational Poynting vector provides a mechanism for the transfer of gravitational energy to a system of falling objects. In the following we will show that the gravitational poynting vector together with the gravitational Larmor theorem also provides a mechanism to explain how massive bodies acquire rotational kinetic energy when external mechanical forces are applied on them.
A relatively recent study by Mars and Senovilla provided us with a uniqueness result for the exterior vacuum gravitational field generated by an isolated distribution of matter in axial rotation in equilibrium in General Relativity. The generalisation to exterior electrovacuum gravitational fields, to include charged rotating objects, is presented here.
On the basis of Lagrangian formalism of relativistic field theory post-Newtonian equations of motion for a rotating body are derived in the frame of Feynman's quantum field gravity theory (FGT) and compared with corresponding geodesic equations in general relativity (GR). It is shown that in FGT the trajectory of a rotating test body does not depend on a choice of a coordinate system. The equation of translational motion of a gyroscope is applied to description of laboratory experiments with free falling rotating bodies and rotating bodies on a balance scale. Post-Newtonian relativistic effect…
New rotation laws have been recently found for general-relativistic self-gravitating stationary fluids. It was not clear whether they apply to systems rotating with a constant linear velocity. In this paper we fill this gap. The answer is positive. That means, in particular, that these systems should exhibit the recently discovered general-relativistic weak-field effects within rotating tori: the dynamic anti-dragging and the deviation from the Keplerian motion induced by the fluid selfgravity.
We prove that, given a stress-free, axially symmetric elastic body, there exists, for sufficiently small values of the gravitational constant and of the angular frequency, a unique stationary axisymmetric solution to the Einstein equations coupled to the equations of relativistic elasticity with the body performing rigid rotations around the symmetry axis at the given angular frequency.
It is shown that Collas and Klein (ArXiv : 0811.2471 [gr-qc]) wrongly concluded that "negative frame dragging" phenomenon takes place at all finite $r$ and $z$ coordinate values . We argue that a test particle with zero angular momentum counter-rotates with respect to the source in the "time machine" region only. In addition, Bonnor's spacetime has an event horizon at $r_{H}$ = 0.
We study two type effects of gravitational field on mechanical gyroscopes (i.e. rotating extended bodies). The first depends on special relativity and equivalence principle. The second is related to the coupling (i.e. a new force) between the spins of mechanical gyroscopes, which would violate the equivalent principle. In order to give a theoretical prediction to the second we suggest a spin-spin coupling model for two mechanical gyroscopes. An upper limit on the coupling strength is then determined by using the observed perihelion precession of the planet's orbits in solar system. We also giv…
This paper computes co-rotating and contra-rotating impact-parameter formulas in the plane of symmetry for any plane symmetric and axisymmetric rotating body in all metric theories of gravity, including general relativity. Impact-parameter formulas are useful to compute the appearance of accreting black holes, neutron stars, and other emitting or reflecting matter near a gravitationally compact rotating body. These rotating-body impact-parameter formulas generalize similar impact-parameter formulas for the Kerr case derived by Bardeen and coworkers in 1972, and another general-metric formula f…
Examples of axially symmetric solutions to Einstein's field equations are given that exhibit anomalous "negative frame dragging" in the sense that zero angular momentum test particles acquire angular velocities in the opposite direction of rotation from the source of the metric.
Recent general-relativistic extensions of Newtonian rotation laws for self-gravitating stationary fluids allow one to rederive, in the first post-Newtonian approximation, the well known geometric dragging of frames, and two new weak-field effects within rotating tori. These are the recently discovered anti-dragging and a new effect that measures the deviation from the Keplerian motion and/or the contribution of the fluids selfgravity. They can be applied to the study of the existence of the (post-)Newtonian limits of solutions and in investigations of inequalities relating parameters of rotati…
Entropy bounds in black hole physics, based on a wide variety of different approaches, have had a long and distinguished history. Recently the current authors have turned attention to uncollapsed systems and obtained a robust entropy bound for uncollapsed static spherically symmetric configurations. In the current article we extend this bound to rotating systems. This extension is less simple than one might at first suppose. Purely classically, (using only classical general relativity and basic thermodynamics), it is possible to show that the entropy of uncollapsed matter inside a region enclo…
Owing to Earth's rotation a free-fall body would move in an elliptical orbit rather than along a straight line forward to the center of the Earth. In this paper on the basis of the theory for spin-spin coupling between macroscopic rotating bodies we study violation of the equivalence principle from long-distance free-fall experiments by means of a rotating ball and a non-rotating sell. For the free-fall time of 40 seconds, the difference between the orbits of the two free-fall bodies is of the order of 10^{-9}cm which could be detected by a SQUID magnetometer owing to such a magnetometer can b…
The problem of describing isolated rotating bodies in equilibrium in General Relativity has so far been treated under the assumption of the circularity condition in the interior of the body. For a fluid without energy flux, this condition implies that the fluid flow moves only along the angular direction, i.e. there is no convection. Using this simplification, some recent studies have provided us with uniqueness and existence results for asymptotically flat vacuum exterior fields given the interior sources. Here, the generalisation of the problem to include general sources is studied. It is pr…
The problem of constructing global models describing isolated axially symmetric rotating bodies in equilibrium is analyzed. It is claimed that, whenever the global spacetime is constructed by giving boundary data on the limiting surface of the body and integrating Einstein's equations both inside and ouside the body, the problem becomes overdetermined. Similarly, when the spacetime describing the interior of the body is explicitly given,the problem of finding the exterior vacuum solution becomes overdetermined. We discuss in detail the procedure to be followed in order to construct the exterio…
In this paper, the dynamical equations and junction conditions at the interface between adjacent layers of different elastic properties for an elastic deformable astronomical body in the first post-Newtonian approximation of Einstein theory of gravity are discussed in both rotating Cartesian coordinates and rotating spherical coordinates. The unperturbed rotating body (the ground state) is described as uniformly rotating, stationary and axisymmetric configuration in an asymptotically flat space-time manifold. Deviations from the equilibrium configuration are described by means of a displacemen…
We formulate new general-relativistic extensions of Newtonian rotation laws for self-gravitating stationary fluids. They have been used to re-derive, in the first post-Newtonian approximation, the well known geometric dragging of frames. We derive two other general-relativistic weak-field effects within rotating tori: the recently discovered dynamic anti-dragging and a new effect that measures the deviation from the Keplerian motion and/or the contribution of the fluids selfgravity. One can use the rotation laws to study the uniqueness and the convergence of the post-Newtonian approximations, …
We present a direct confirmation of the validity of the equivalence principle for unstructured test bodies in scalar tensor gravity. Our analysis is complementary to previous approaches and valid for a large class of scalar-tensor theories of gravitation. A covariant approach is used to derive the equations of motion in a systematic way and allows for the experimental test of scalar-tensor theories by means of extended test bodies.
We propose the almost-geodesic motion of self-gravitating test bodies as a possible selection rule among metric theories of gravity. Starting from a heuristic statement, the "gravitational weak equivalence principle", we build a formal, operative test able to probe the validity of the principle for any metric theory of gravity, in an arbitrary number of spacetime dimensions. We show that, if the theory admits a well-posed variational formulation, this test singles out only the purely metric theories of gravity. This conclusion reproduces known results in the cases of general relativity (also w…
We perform some numerical study of the secular triaxial instability of rigidly rotating homogeneous fluid bodies in general relativity. In the Newtonian limit, this instability arises at the bifurcation point between the Maclaurin and Jacobi sequences. It can be driven in astrophysical systems by viscous dissipation. We locate the onset of instability along several constant baryon mass sequences of uniformly rotating axisymmetric bodies for compaction parameter $M/R = 0-0.275$. We find that general relativity weakens the Jacobi like bar mode instability, but the stabilizing effect is not very …
Using post-Newtonian equations of motion for fluid bodies valid to the second post-Newtonian order, we derive the equations of motion for binary systems with finite-sized, non-spinning but arbitrarily shaped bodies. In particular we study the contributions of the internal structure of the bodies (such as self-gravity) that would diverge if the size of the bodies were to shrink to zero. Using a set of virial relations accurate to the first post-Newtonian order that reflect the stationarity of each body, and redefining the masses to include 1PN and 2PN self-gravity terms, we demonstrate the comp…
The dynamics of spinning test bodies, moving in rotating black hole (Kerr, Bardeen-like and Hayward-like) spacetimes, are investigated. In Kerr spacetime, all the spherical, zoom-whirl and unbound orbits are considered numerically. Along spherical orbits and for high spin, an amplitude modulation is found in the harmonic evolution of the spin precessional angular velocity, caused by the spin-curvature coupling. Along the discussed zoom-whirl and unbound orbits, the test body approaches the center so much that it passes through the ergosphere. Near and inside the ergosphere, the variation of th…
The differential acceleration between a rotating mechanical gyroscope and a non-rotating one is directly measured by using a double free-fall interferometer, and no apparent differential acceleration has been observed at the relative level of 2x10{-6}. It means that the equivalence principle is still valid for rotating extended bodies, i.e., the spin-gravity interaction between the extended bodies has not been observed at this level. Also, to the limit of our experimental sensitivity, there is no observed asymmetrical effect or anti-gravity of the rotating gyroscopes as reported by hayasaka et…
Recent interest on studying possible violations of the Equivalence Principle has led to the development of space satellite missions testing it for bodies moving on circular orbits around Earth. This experiment establishes that the validity of the Equivalence Principle is independent of the composition of bodies. However, the internal degrees of freedom of the bodies (such as spin) were not taken into account. In this work, it is shown exactly that the circular orbit motion of test bodies does present a departure from geodesic motion when spin effects are not negligible. Using a Lagrangian theo…
Formulation of the Lagrangian approach is presented for studying features of motions of stellar bodies with non-zero rest mass in the vicinity of fast-spinning black holes. The structure of Lagrangian is discussed. The general method is applied to description of body motion in the Kerr model of space-time to transition to the problem of tidal disruption of elastic bodies by strong gravitational field.
Quasi-stationary (i.e. parametric) transitions from rotating equilibrium configurations of fluid bodies to rotating black holes are discussed. For the idealized model of a rotating disc of dust, analytical results derived by means of the "inverse scattering method" are available. They are generalized by numerical results for rotating fluid rings with various equations of state. It can be shown rigorously that a black hole limit of a fluid body in equilibrium occurs if and only if the gravitational mass becomes equal to twice the product of angular velocity and angular momentum. Therefore, any …
A space with deformed commutation relations for coordinates and momenta leading to generalized uncertainty principle (GUP) is studied. We show that GUP causes great violation of the weak equivalence principle for macroscopic bodies, violation of additivity property of the kinetic energy, dependence of the kinetic energy on composition, great corrections to the kinetic energy of macroscopic bodies. We find that all these problems can be solved in the case of arbitrary deformation function depending on momentum if parameter of deformation is proportional inversely to squared mass.
The Einstein's equivalence principle and experiments, in which bodies and observers are in different G potentials, have been used to prove that the chain of hypothesis coming from assuming the absolute invariability of the bodies, after a change of G potential, are wrong. The absolute changes of frequencies, masses and lengths of every well-defined part of the non-local bodies, compared with the local ones, are linearly related to the differences of G potential between the non-local bodies and the observer. Such absolute changes are independent on the forces within the structure of the bodies.…
We used a continuously rotating torsion balance instrument to measure the acceleration difference of beryllium and titanium test bodies towards sources at a variety of distances. Our result Delta a=(0.6+/-3.1)x10^-15 m/s^2 improves limits on equivalence-principle violations with ranges from 1 m to infinity by an order of magnitude. The Eoetvoes parameter is eta=(0.3+/-1.8)x10^-13. By analyzing our data for accelerations towards the center of the Milky Way we find equal attractions of Be and Ti towards galactic dark matter, yielding eta=(-4 +/- 7)x10^-5. Space-fixed differential accelerations i…
We propose the metric for general rotating spacetimes. These spacetimes are stationary, axially symmetric and spatially asymptotically flat. They can be the spacetimes outside of rotating black holes or rotating celestial bodies such as the Sun and the Earth. The metric functions are expanded in power series of distance and the angle variable is included in the expansion coefficients.
We present an extension to a previous work to study the collapse of a radiating, slow-rotating self-gravitating relativistic configuration. In order to simulate dissipation effects due to the transfer of photons and/or neutrinos within the matter configuration, we introduce the flux factor, the variable Eddington factor and a closure relation between them. Rotation in General Relativity is considered in the slow rotation approximation, i.e. tangential velocity of every fluid element is much less than the speed of light and the centrifugal forces are little compared with the gravitational ones.…
A general principle of non-equivalence for bodies and observers in different G potentials (GP) was derived from correspondence of the Einstein's equivalence principle either with optical physics or with gravitational experiments in which bodies and observers are in different GP. According to it some relative physical changes occur to any well defined part of an object after a change of GP.. Such changes cannot be measured by observers travelling with the object because his instruments change in identical proportions. The same principle was derived from a new gravitational theory based on a par…
We study neutrino oscillations in a rotating spacetime under the weak gravity limit for the trajectories of neutrinos which are constrained in the equatorial plane. Using the asymptotic form of the Kerr metric, we show that the rotation of the gravitational source non-trivially modifies the neutrino phase. We find that the oscillation probabilities deviate significantly from the corresponding results in the Schwarzschild spacetime when neutrinos are produced near the black hole (still in the weak-gravity limit) with non-zero angular momentum and detected on the same side, i.e., the non-lensed …
Studies in string theory and quantum gravity lead to the Generalized Uncertainty Principle (GUP) and suggest the existence of a fundamental minimal length which, as was established, can be obtained within the deformed Heisenberg algebra. The first look on the classical motion of bodies in a space with corresponding deformed Poisson brackets in a uniform gravitational field can give an impression that bodies of different mass fall in different ways and thus the equivalence principle is violated. Analyzing the kinetic energy of a composite body we find that the motion of its center of mass in th…
We study the coupling of massive scalar fields to matter in orbit around rotating black holes. It is generally expected that orbiting bodies will lose energy in gravitational waves, slowly inspiralling into the black hole. Instead, we show that the coupling of the field to matter leads to a surprising effect: because of superradiance, matter can hover into "floating orbits" for which the net gravitational energy loss at infinity is entirely provided by the black hole's rotational energy. Orbiting bodies remain floating until they extract sufficient angular momentum from the black hole, or unti…
We give a physical interpretation to the multi-polar Erez-Rozen-Quevedo solution of the Einstein Equations in terms of bars. We find that each multi-pole correspond to the Newtonian potential of a bar with linear density proportional to a Legendre Polynomial. We use this fact to find an integral representation of the $γ$ function. These integral representations are used in the context of the inverse scattering method to find solutions associated to one or more rotating bodies each one with their own multi-polar structure.
In massive gravity, galileon, and braneworld explanations of cosmic acceleration, force modifications are screened by nonlinear derivative self-interactions of the scalar field mediating that force. Interactions between the field of a central body ("A") and an orbiting body ("B") imply that body B does not move as a test body in the field of body A if the orbit is smaller than the Vainshtein radius of body B. We find through numerical solutions of the joint field at the position of B that the A-field Laplacian is nearly perfectly screened by the B self-field, whereas first derivative or net fo…
Most theories that predict time and/or space variation of fundamental constants also predict violations of the Weak Equivalence Principle. In 2004 Khoury and Weltman proposed the so called chameleon field arguing that it could help avoiding experimental bounds on the WEP while having a non-trivial cosmological impact. In this paper we revisit the extent to which these expectations continue to hold as we enter the regime of high precision tests. The basis of the study is the development of a new method for computing the force between two massive bodies induced by the chameleon field which takes…
The effective action for QED in curved spacetime includes equivalence principle violating interactions between the electromagnetic field and the spacetime curvature. These interactions admit the possibility of superluminal yet causal photon propagation in gravitational fields. In this paper, we extend our analysis of photon propagation in gravitational backgrounds to the Kerr spacetime describing a rotating black hole. The results support two general theorems -- a polarisation sum rule and a `horizon theorem'. The implications for the stationary limit surface bounding the ergosphere are also d…
Location dependence of physical parameters such as the electromagnetic fine structure constant and Newton's G produce body accelerations which violate universality of free fall rates testable with laboratory and space experiments. Theoretically related cosmological time variation of these same parameters are also constrained by experiments such as lunar laser ranging, and these time variations produce accelerations of bodies relative to a preferred cosmological frame.
The Ricci Rotation Coefficients in the description of trajectories of spinning test particles off-equatorial planes in a rotational gravitational field
On incorporating special relativity theory into an extended equivalence principle, post-Newtonian gravitational phenomena beyond that originally predicted by Einstein are predicted (required), such as geodetic and gravitomagnetic precessions of local inertial frames, and precession of Mercury's orbital perihelion. Why were not these phenomena predicted in the years 1907-1911? The unique 1/c^2 order dynamical equations for clock rates and motion of both bodies and light in local gravity are derived which guarantee fulfillment of the special relativistic equivalence principle.
In most studies of equivalence principle violation by solar system bodies, it is assumed that the ratio of gravitational to inertial mass for a given body deviates from unity by a parameter Delta which is proportional to its gravitational self-energy. Here we inquire what experimental constraints can be set on Delta for various solar system objects when this assumption is relaxed. Extending an analysis originally due to Nordtvedt, we obtain upper limits on linearly independent combinations of Delta for two or more bodies from Kepler's third law, the position of Lagrange libration points, and t…
Using a multi domain spectral method, we investigate systematically the general-relativistic model for axisymmetric uniformly rotating, homogeneous fluid bodies generalizing the analytically known Maclaurin and Schwarzschild solutions. Apart from the curves associated with these solutions and a further curve of configurations that rotate at the mass shedding limit, two more curves are found to border the corresponding two parameter set of solutions. One of them is a Newtonian lens shaped sequence bifurcating from the Maclaurin spheroid sequence, while the other one corresponds to highly relati…
A highly accurate, multi-domain spectral code is used in order to construct sequences of general relativistic, differentially rotating neutron stars in axisymmetry and stationarity. For bodies with a spheroidal topology and a homogeneous or an N=1 polytropic equation of state, we investigate the solution space corresponding to broad ranges of degree of differential rotation and stellar densities. In particular, starting from static and spherical configurations, we analyse the changes of the corresponding surface shapes as the rate of rotation is increased. For a sufficiently weak degree of dif…
The motion of an extended body up to the quadrupolar structure is studied in the Schwarzschild background following Dixon's model and within certain restrictions (constant frame components for the spin and the quadrupole tensor, center of mass moving along a circular orbit, etc.). We find a number of interesting situations in which deviations from the geodesic motion, due to the internal structure of the particle, can originate measurable effects. However, the standard clock-effect for a pair co/counter-rotating bodies spinning up/down is not modified by the quadrupolar structure of the partic…
Maintaining the relativity of all motion, especially rotational motion, Mach denied the existence of absolute motion and absolute space. He maintained the equivalence of the Ptolemaic and the Copernican systems and the equivalence of a fixed bucket in a rotating universe with the converse. An analysis of the Foucault pendulum shows that there cannot be a fixed bucket in a rotating universe. Also, Mach's views violate the physics he espoused: non-inertial experiments, e.g. stellar aberration and electromagnetic effects, distinguish between a rotating bucket in a fixed universe and the converse,…
We study the motion of test particles in the gravitational field of a rotating and deformed object within the framework of the adiabatic theory. For this purpose, the Hartle-Thorne metric written in harmonic coordinates is employed in the post-Newtonian approximation where the adiabatic theory is valid. As a result, we obtain the perihelion shift formula for test particles orbiting on the equatorial plane of a rotating and deformed object. Based on the perihelion shift expression, we show that the principle of superposition is valid for the individual effects of the gravitational source mass, …
