Reference Frames and Gravitomagnetism

The following sections are included:

• PREFACE

• CONTENTS

A review of spacetime splitting techniques and gravitoelectromagnetism is presented.

The definition of inertial forces in general relativity is discussed and then illustrated for circular orbits in black hole spacetimes.

After a brief introduction on de Sitter precession, gravitomagnetism and Lense-Thirring effect, we propose an invariant characterization of Lense-Thirring effect and gravitomagnetism to distinguish these phenomena from the effects arising from the motion of a test particle on a static background such as the de Sitter effect.

First, we briefly describe some of the general relativistic, gravitomagnetic, phenomena arising in the vicinities of a spinning body, due to its rotation, and some of the historical attempts to detect and measure gravitomagnetism and Lense-Thirring effect, including the LARES experiment, a phase-A space mission to measure "frame-dragging" with accuracy of less than 3% and to provide other basic tests of general relativity and gravitation. We then describe the method to measure the Lense-Thirring effect by analyzing the orbits of the two laser-ranged satellites LAGEOS and LAGEOS II; this method has provided the direct measurement of Earth's gravitomagnetism. We report on these 1995-2000 experimental evidences of the Lense-Thirring effect obtained by analyzing the nodes of LAGEOS and LAGEOS II and the perigee of LAGEOS II with the orbital programs GEODYN-SOLVE, using the Earth's Gravitational Models JGM-3 and EGM-96, and this new method. The first detection was obtained in 1995, the most accurate measurements were obtained in 1998-2000 using EGM-96, with accuracy of the order of 20%. Finally, we present our new, preliminary, result obtained by analyzing 5 years of data of the LAGEOS satellites.

Basic elements for a relativistic theory of coordinate systems are introduced. The main purposes of such a theory are to precise the physical and geometrical status of coordinate systems in general relativity, to structure those presently known, to offer a convenient scheme to incorporate new ones, to reveal voids in our knowledge of their mutual relations, and to incitate their study. Relativistic operational criteria to construct coordinate systems are given, with particular attention to satellite positioning systems, to which the current GPS could be related.

We consider the formation of stable non-Keplerian rings of matter around rotating black holes. These rings can accomodate particles with energy much higher than what would be allowed on stable Keplerian orbits and in some cases they can "stably" store an infinite (in principle) amount of energy. Here we show that this particular phenomenon, is a consequence of a general relativistic effect which states that non geodesic circular orbits with maximal or minimal acceleration occur, in the Kerr metric, with a non zero angular velocity with respect to infinity. From the astrophysical point of view, a sudden release of the large amount energy which may have been confined on those rings, can account for the rapid variability observed in most of the active galactic nuclei.

Recent modern space missions deliver invaluable information about origin of our universe, physical processes in the vicinity of black holes and other exotic astrophysical objects, stellar dynamics of our galaxy, etc. On the other hand, space astrometric missions make it possible to determine with unparalleled precision distances to stars and cosmological objects as well as their physical characteristics and positions on the celestial sphere. Permanently growing accuracy of space astronomical observations and the urgent need for adequate data processing algorithms require corresponding development of an adequate theory of reference frames along with unambiguous description of propagation of light rays from a source of light to observer. Such a theory must be based on the Einstein's general relativity and account for numerous relativistic effects both in the solar system and outside of its boundary. The main features of the relativistic theory of reference frames are presented in this work. A hierarchy of the frames is described starting from the perturbed cosmological Friedmann-Robertson-Walker metric and going to the observer's frame through the intermediate barycentric and geocentric frames in the solar system. Microarcsecond astrometry and effects of propagation of light rays in time-dependent gravitational fields are discussed as well.

Recent developments in string theory have led to 5-dimensional warped spacetime models in which standard-model fields are confined to a 3-brane (the observed universe), while gravity can propagate in the fifth dimension. Gravity is localized near the brane at low energies, even if the extra dimension is noncompact. A review is given of the classical geometry and dynamics of these brane-world models. The field equations on the brane modify the general relativity equations in two ways: local 5-D effects are imprinted on the brane as a result of its embedding, and are significant at high energies; nonlocal effects arise from the 5-D Weyl tensor. The Weyl tensor transmits tidal (Coulomb), gravitomagnetic and gravitational wave effects to the brane from the 5-D nonlocal gravitational field. Local high-energy effects modify the dynamics of inflation, and increase the amplitude of scalar and tensor perturbations generated by inflation. Nonlocal effects introduce new features in cosmological perturbations. They induce a non-adiabatic mode in scalar perturbations and massive modes in vector and tensor perturbations, and they can support vector perturbations even in the absence of matter vorticity. In astrophysics, local and nonlocal effects introduce fundamental changes to gravitational collapse and black hole solutions.

Gravitoelectromagnetism is briefly reviewed and some recent developments in this topic are discussed. The stress-energy content of the gravitoelectromagnetic field is described from different standpoints. In particular, the gravitational Poynting flux is analyzed and it is shown that there exists a steady flow of gravitational energy circulating around a rotating mass.

The basic physical structure of the relativistic theory of gravitation is discussed. The significant role that the Hypothesis of Locality plays in relativity theory is elucidated via the phenomenon of spin-rotation coupling. The limitations of this hypothesis are critically examined. A nonlocal theory of accelerated observers is presented and some of its observational consequences are described.

We show how to generalize the classical electric-magnetic decomposition of the Maxwell or the Weyl tensors to arbitrary fields described by tensors of any rank in general n-dimensional spacetimes of Lorentzian signature. The properties and applications of this decomposition are reviewed. In particular, the definition of tensors quadratic in the original fields and with important positivity properties is given. These tensors are usually called "super-energy" (s-e) tensors, they include the traditional energy-momentum, Bel and Bel-Robinson tensors, and satisfy the so-called Dominant Property, which is a straightforward generalization of the classical dominant energy condition satisfied by well-behaved energy-momentum tensors. We prove that, in fact, any tensor satisfying the dominant property can be decomposed as a finite sum of the s-e tensors. Some remarks about the conservation laws derivable from s-e tensors, with some explicit examples, are presented. Finally, we will show how our results can be used to provide adequate generalizations of the Rainich conditions in general dimension and for any physical field.

We show that any tensor satisfying the dominant energy condition on an N-dimensional Lorentzian manifold can be written as a sum of N-1 superenergy tensors of simple forms and the metric. We also prove that the set of non-singular superenergy tensors of simple forms is precisely the set of tensors proportional to involutary time orientation preserving Lorentz transformations. Finally, we find generalised algebraic Rainich conditions, i.e. ways of determining the physics from the geometry, for energy-momentum tensors of arbitrary trace in arbitrary dimension.

It is said that the vorticity of a congruence plays the role of rate of rotation for the precession of a gyroscope moving along a world-line belonging to the congruence. Our aim is to determine the evolution equation for the angular momentum of a gyroscope with respect to an arbitrary time-like congruence, i.e, a reference congruence which does not contain the curve described by the gyroscope. In particular, we will show what conditions are needed to support the previous assert about the vorticity. So, we establish a well-founded theoretical description for the analysis of the precession of gyroscopes, providing suitable conclusions for an eventual performance of planned experiments.

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.

The Bel tensor is divergence-free in some important cases leading to the existence of conserved currents associated to Killing vectors analogously to those of the energy-momentum tensor. When the divergence of the Bel tensor does not vanish one can study the interchange of some quantities between the gravitational and other fields obtaining mixed total conserved currents. Nevertheless, the Bel currents are shown to be conserved (independently of the matter content) if the Killing vectors satisfy some very general conditions. These properties are similar to some very well known statements for the energy-momentum tensor.

New techniques to evaluate the clock effect are described. They are based on the flatness of the cylindrical surface containing the world lines of the rays constrained to move on circular trajectories about a spinning mass. The effect of the angular momentum of the source is manifested in the fact that inertial observers must be replaced by local non rotating observers. Starting from this, exact formulas for circular trajectories are found. Numerical estimates for the Earth environment show that light would be a better probe than actual clocks to evidence the angular momentum influence.

No abstract received.

The conditions for thermal equilibrium in a given space-time (i.e. given metric) are derived from the assumption that the entropy per baryon is a maximum. Equilibrium of a star is studied in two steps. Firstly local equilibrium in a given space-time is derived from the condition that the entropy per baryon is a maximum for fixed baryon density. Examples of classical and Fermi ideal gases are considered. Then the results are applied to the study of global equilibrium configurations of non-rotating spherical stars.

We propose a new definition of entropy for any mass m, based on gravitation and through the concept of a gravitational cross section. It turns out to be proportional to mass, and therefore extensive, and to the age of the Universe. It is a Machian approach. It is also the number of gravity quanta the mass has emitted through its age. The entropy of the Universe is so determined and the cosmological entropy problem solved.

In a recent paper Carot et al. considered the definition of cylindrical symmetry as a specialisation of the case of axial symmetry. One of their propositions states that if there is a second Killing vector, which together with the one generating the axial symmetry, forms the basis of a two-dimensional Lie algebra, then the two Killing vectors must commute, thus generating an Abelian group. In this paper a similar result, valid under considerably weaker assumptions, is derived: any two-dimensional Lie transformation group which contains a one-dimensional subgroup whose orbits are circles, must be Abelian. The method used to prove this result is extended to apply to three-dimensional Lie transformation groups. It is shown that the existence of a one-dimensional subgroup with closed orbits restricts the Bianchi type of the associated Lie algebra to be I, II, III, VII_{q = 0}, VIII or IX. Some results on n-dimensional Lie groups are also derived and applied to show there are severe restrictions on the structure of the allowed four-dimensional Lie transformation groups compatible with cyclic symmetry.

We present a flat (K = 0) cosmological model, described by a perfect fluid with the "constants" G, c and Λ varying with cosmological time t. We introduce Planck's "constant" ħ in the field equations through the equation of state for the energy density of radiation. We then determine the behaviour of the "constants" by using the zero divergence of the second member of the modified Einstein's field equations i.e. , together with the equation of state and the Einstein cosmological equations. Assuming realistic physical and mathematical conditions we obtain a consistent result with ħc = constant. In this way we obtain gauge invariance for the Schrödinger equation and the behaviour of the remaining "constants".

We study lightlike distributions of codimension 1 for any observer U and we define the U-parallelizations that describe these distributions. We obtain two kinds of representations: Ω and . These distributions describe the two different directions of propagation of the system, always respect to the observer U. Relations between different observers respect to a given lightlike distribution are studied, and necessary conditions for the integrability of Ω and are given. Finally, we show by a counterexample that if Ω is a foliation then is not necessarily a foliation. In this case, the observer U is not a suitable observer.

Scalar field spacetimes are considered with a view towards their applications in cosmology. Some results existing in the literature are reviewed and some new others are proven concerning inhomogeneous G₂ cosmological models. Dynamical systems theory is used to analyze the qualitative behavior of certain families of solutions.

In this proceeding we would like to discuss the Thermodynamics together with the Dominant Energy Condition (DEC) for different theories of gravity in Kasner type cosmological solutions.

We describe the derivation of the Einstein equations for twisting type-N gravitational fields with a group of symmetries H₂, as well as a generalization of a procedure for reducing the equations to a single third-order real ordinary differential equation, developed by one of us for the case with two Killing vectors. We discuss the case ɸ = -1, where ɸ is the homothetic parameter.

The Stephani Universes that can be interpreted as an ideal gas evolving in local thermal equilibrium are determined, and the method to obtain the associated thermodynamic schemes is given.

Our aim is to give a general framework for the collapse of spherically symmetric objects and for voids. To accomplish this we will consider the matching of two spherically symmetric space-time regions. We will give a set of conditions to ensure us that the space-time has a physical meaning. Finally we will be able to present a set of necessary conditions which allow us to ascertain which matchings are feasible and which are not allowed. We shall illustrate these results by applying them to reject some families of models and by showing particular well-behaved ones.

In vacuum space-times the exterior derivative of a Killing vector field is a two-form that satisfies Maxwell equations without electromagnetic sources. Using the algebraic structure of this two-form we have set up a new formalism for the study of vacuum space-times with an isometry.

In this talk relativistic corrections due to Geroch-Hansen multipoles for perihelion precession and node line precession of orbits in a stationary axially symmetric vacuum spacetime endowed with a plane of symmetry will be shown. Patterns of regularity will be discussed.

In this paper we determine all the congruences υ satisfying the equation *W(υ; υ) = 0 for every Petrov type of the Weyl tensor, and study the causal character of the solutions.

The quasi-normal mode (QNM) ring-down of massive black holes excited at their birth in the gravitational collapse of stellar configurations may be detectable by the new generation of gravitational wave interferometers. We study new qualitative features emerging in the emitted QNM signal in the presence of accretion events which would add a significant fraction of mass to the central black hole on a dynamical timescale. To this aim we numerically simulate spherically symmetric general relativistic black hole accretion of perfect fluid matter in combination with appropriate modeling of non-spherical spacetime perturbations of the underlying solution. It is shown that the QNM frequency, dimensionally scaling with the mass of the black hole, is a sensitive indicator of the mass accretion rate.

Using a two-fluid approach we write the equations describing the evolution of density and magnetic field inhomogeneities and curvature perturbations in a matter-radiation universe as a fourth order autonomous dynamical system. Analysis of the equilibrium points for the radiation dominated era lead to solutions similar to the super-horizon modes found analytically by Tsagas and Maartens. We find that a study of the dynamical system in the dust-dominated era leads naturally to a magnetic critical length scale closely related to the Jeans Length. Depending on the size of wavelengths relative to this scale, these solutions show three distinct behaviours: large-scale stable growing modes, intermediate decaying modes, and small-scale damped oscillatory solutions.

In the Friedmann universe the scale of length is a function of time a ∝ t^α. From this it follows that the Newton potential introduced in the local region of the universe is a function of time Φ ∝ t^-α. Thus one comes to the contradiction between the Friedmann universe and the local physics. To maintain the Newton gravity in the local region it is necessary to introduce the law of evolution for the scale of mass the same as that for the scale of length m ∝ t^α. The model of the universe is developed in this way.

The asymptotic properties of the solutions to the Einstein-Maxwell equations with boost-rotation symmetry and Petrov type D are studied. We find series solutions to the pertinent set of equations which are suitable for a late time descriptions in coordinates which are well adapted for the description of the radiative properties of spacetimes (Bondi coordinates). By calculating the total charge, Bondi and NUT mass and the Newman-Penrose constants of the spacetimes we provide a physical interpretation of the free parameters of the solutions. Additional relevant aspects on the asymptotics and radiative properties of the spacetimes considered, such as the possible polarization states of the gravitational and electromagnetic field, are discussed through the way.

Different concepts of time dilation are analized and critically compared. a) Classical time dilation, as a consequence of the independence of electromagnetic waves velocity from the velocity of their source. b) Time dilation in Lorentz-Poincaré's theory of relativity, included in the apparent equivalence of inertial frames, as a consequence of Maxwell's equations. c) Time dilation in Einstein's theory of relativity, as a consequence of a real equivalence of inertial frames.

The theory of General Relativity explaines the advance of Mercury perihelion using space curvature and the Schwarzschild metric. We demonstrate that this phenomena can also be interpreted due to the cogravitational field produced by the apparent motion of the Sun around Mercury giving exactly the same estimate as derived from the Schwarzschild metric in general relativity theory. This is a surprising and new result because the estimate from both theoretical approaches match exactly the measured value. The discussion and implications of this result is out of the scope of the present work.

The Gertsenshtein effect, discovered in 1962, produced some work, in the seventies, about the possibility of generating, and detecting, high frequency gravitational waves. Unfourtunately, the efficiency of the generator imagined, a cavity resonator, was too small, many orders bellow the observability threshold. In this contribution we discuss the foundation of a wave generator that could produce gravitational waves with amplitude major than the detection's threshold.

We give a new presentation of the Petrov classification of Einstein spaces. In our approach, the space of bivectors Λ₂, which underlies the double 2-form nature of the Riemann tensor, is treated as subspace of the whole Grassmann space Λ, on which the Clifford geometric algebra is naturally defined. We use this product to make explicit the intrinsic complex structure that bivector space contains, without introducing any imaginary unit i.

We study the generation of gravitational radiation by sources moving in the de Sitter background. A gauge is chosen in which all the physical and unphysical modes of the graviton correspond to minimally coupled massless scalar fields in de Sitter space-time and massless scalar fields in Minkowski space-time. The graviton retarded Green's function is obtained in this gauge using quantum field theory techniques. We obtain closed formulas for the spectral decomposition in frequencies of the linearized gravitational field produced by the source in terms of a suitable spectral decomposition of the source energy-momentum tensor.

The following sections are included:

• LIST OF PARTICIPANTS