Statefinders and observational measurement of superenergy

The superenergy of the universe is a tensorial quantity and it is a general relativistic analogue of the Appell's energy of acceleration in classical mechanics. We propose the way to measure this quantity by the application of the observational parameters such as the Hubble parameter, the deceleration parameter, the jerk and the snap (kerk), known as statefinders. We show that the superenergy of gravity requires only the Hubble and deceleration parameters to be measured, while the superenergy of matter requires also the measurement of the higher-order characteristics of expansion: the jerk and the snap. In such a way, the superenergy becomes another parameter characterizing the evolution of the universe. One of the interesting points is that the cosmological constant has a purely gravitational interpretation in terms of superenergy.     

Positivity of General Superenergy Tensors

Several of the most important results in general relativity require or assume positivity properties of certain tensors. The positive energy theorem and the singularity theorems make assumptions about the energy-momentum tensor and Ricci tensor respectively. Positivity of the Bel–Robinson tensor is needed in the proof of the global stability of Minkowski spacetime. Senovilla has recently presented a procedure of how to construct a superenergy tensor from any tensor. For a Maxwell field or a scalar field the procedure yields the usual energy-momentum tensor, for the Weyl tensor and the Riemann tensor one obtains the Bel–Robinson tensor and Bel tensor respectively. In general, by considering any tensor as an r-fold n ₁,…,n _r )-form, one constructs a rank 2r superenergy tensor from it. By using spinor methods, we prove that the contraction of any such superenergy tensor with 2r future-pointing vectors is non-negative. We refer to this as the dominant superenergy property and it generalizes several previous positivity results obtained for certain tensors as well as it provides a unified way of treating them. Some more examples are given and applications discussed. Received: 21 December 1998 / Accepted: 5 May 1999     

Sparling Two-Forms, the Conformal Factor and the Gravitational Energy Density of the Teleparallel Equivalent of General Relativity

It has been shown recently that within the framework of the teleparallel equivalent of general relativity (TEGR) it is possible to define the energy density of the gravitational field in a unique way. The tegr amounts to an alternative formulation of Einstein's general relativity, not to an alternative gravity theory. The localizability of the gravitational energy has been investigated in a number of spacetimes with distinct topologies, and the outcome of these analyses agree with previously known results regarding the exact expression of the gravitational energy, and/or with the specific properties of the spacetime manifold. In this article we establish a relationship between the expression of the gravitational energy density of the TEGR and the Sparling two-forms, which are known to be closely connected with the gravitational energy. We will also show that our expression of energy yields the correct value of gravitational mass contained in the conformal factor of the metric field.     

Weak Lensing, Shear and the Cosmic Virial Theorem in a Model with a Scale-Dependent Gravitational Coupling

It is argued that, in models where the gravitational coupling is scaledependent, predictions concerning weak gravitational lensing and shear are essentially similar to the ones derived from General Relativity. This is consistent with recent negative results of observations of the MS1224, CL2218 and A1689 systems aimimg to infer from those methods the presence of dark matter. It is shown, however, that the situation is quite different when an analysis based on the Cosmic Virial Theorem is concerned.     

The Dynamical Behaviour of Test Particles in?a?Quasi-spherical Spacetime and the?Physical Meaning?of?Superenergy

We calculate the instantaneous proper radial acceleration of test particles (as measured by a locally defined Lorentzian observer) in a Weyl spacetime, close to the horizon. As expected from the Israel theorem, there appear some bifurcations with respect to the spherically symmetric case (Schwarzschild) which are explained in terms of the behaviour of the superenergy, bringing out the physical relevance of this quantity in the study of general relativistic systems.     

Letter: Gravitational Entropy of Constrained Instanton

The seeds for quantum creations of universes areconstrained gravitational instantons. For all compactconstrained instantons with a U(1) isometry, the period of the group parameter is identifiedas the reciprocal of the temperature. If remains a free parameter under the constraints, then theEuclidean action becomes the negative of the entropy. Asexamples, we perform the calculations forthe Taub-nut and Taub-Bolt-type models andstudy the quantum creation of the Taub-nutuniverse.     

Canonical superenergy tensors in general relativity

We present the concept of superenergy tensors in the framework of general relativity (GR). These tensors were introduced constructively by the author years ago and they were obtained by a suitable averaging of the energy-momentum tensors or pseudotensors. Because in GR the Einstein canonical energy-momentum pseudotensor_E t _i ^k of the gravitational field and the canonical energy-momentum complex , matter and gravitation,are physically distinguished, we confine this paper to thecanonical superenergy tensor ^g S _i ^k of the gravitational field Γ _kl ⁱ and to the canonical total superenergy tensorS _i ^k = ^g S _i ^k + ^m S _i ^k of matter and gravitation only. These superenergy tensors can be obtained by the above-mentioned averaging of the pseudotensor_E t _i ^k and complex_E K _i ^k . We give the analytic forms of these two canonical superenergy tensors and show some of their possible applications in GR. The canonical superenergy tensor ^g S _i ^k of the gravitational field Γ _kl ⁱ can be used as asubstitute for the nonexisting energy-momentum tensor of this field.     

Some Properties of the Bel and Bel-Robinson Tensors

The properties of the Bel and Bel-Robinson tensors seem to indicate that they are closely related to the gravitational energy-momentum. We present some new properties of these tensors which might throw some light onto this relationship. First, for any spacetime we find a decomposition of the Bel tensor in terms of the Bel-Robinson tensor and two other tensors, which we call the pure matter super-energy tensor and the matter-gravity coupling super-energy tensor. We show that the pure matter super-energy tensor of any Einstein-Maxwell field is simply the square of the usual energy-momentum tensor. This, together with the fact that the Bel-Robinson tensor has dimensions of energy density square, leads us to the definition of square root for the Bel-Robinson tensor: a two-covariant symmetric traceless tensor with dimensions of energy density and such that its square gives the Bel-Robinson tensor. We prove that this square root exists if and only if the spacetime is of Petrov type O, N or D, and its general expression is explicitly presented. The properties of this new tensor are examined and some interesting explicit examples are analyzed. Of particular interest are an invariant function that appears in the spherically symmetric metrics and an expression for the energy carried out by pure plane gravitational waves. We also examine the decomposition of the whole Bel tensor for Vaidya's radiating metric and Kerr-Newman's solution. Finally, we generalize the definition of square root to a factorization of the Bel-Robinson tensor and get the general solution for all Petrov types.     

Radiation and vorticity: the missing link

In the context of general relativity, radiation, either gravitational or electromagnetic, is closely associated to vorticity of observers world lines. We stress in this letter that the factor that relates the two phenomena is a circular flow of energy (electromagnetic) and/or superenergy on the planes orthogonal to vorticity vector. We also stress the potential relevance of the abovementioned relationship in experiments to detect gravitational radiation.     

A Comment on Differential Identities for the Weyl Spinor Perturbations

It is shown that in a type-D vacuum space-time with cosmological constant, the components of the Weyl spinor perturbations along the principal spinors of the background conformal curvature satisfy differential identities, which are valid in all the normalized spin frames {o ^A , ^A } such that o ^A and ^A are double principal spinors of the background conformal curvature.     

Do gravitational waves carry energy‐momentum and angular momentum?

We show that gravitational waves which possess a non‐vanishing Riemann tensor R_iklm ≠ 0 always carry energy‐momentum and angular momentum. Our proof uses canonical superenergy and supermomentum tensors for the gravitational field.     

Gravitational Waves and Discontinuous Motions

The mathematical theory of gravitational wave-fronts is revisited with the help of orthogonal decomposition techniques. Thus many important well known results about gravitational waves in empty space are readily obtained in the local geometry framework and in general coordinates. Then different kinds of motion in the presence of a wave are investigated. Our study shows that a discontinuity effect, in traversing the wave-front, arises in the motion. The importance of this effect increases with the complexity of the considered particle. No essential discontinuity is present in simple motions such as that of a point particle and, differently from what usually believed, that of a dust of particles governed by the geodesic deviation equation.     

The Investigation of the Model of Gravitational Repulsion in Einstein's General Theory of Relativity

The gravitational field of a static, sphericallysymmetric source of mass M and scalar charge q isconsidered. It is shown that the metric expression forthis source is considerably simplified in two limiting cases: a) for M2 4q²/G,that is when the mass of the source is the maincontributor in the gravitational field; b) forq² M²G/4, when theenergy-momentum tensor of the static, spherically symmetric scalar field is the main contributorin the gravitational field. In the limiting caseq² M²G/4, the geodesicsof the massive and massless particles are studied. It isshown that gravitational forces of repulsion act on a particle movingnon-radially in this field. As a result, voids should becreated in the region surrounding such sources in theUniverse. Moreover, the stars with considerable scalar charge q² M²G/4 will act not as convexgravitational lenses as in the case whenq² M²G/4, but as concavegravitational lenses for the electromagnetic rays oflarge impact parameter.     

The Tensors of the Averaged Relative Energy–Momentum and Angular Momentum in General Relativity and Some of Their Applications

There exist different kinds of averaging of the differences of the energy–momentum and angular momentum in normal coordinates NC(P) which give tensorial quantities. The obtained averaged quantities are equivalent mathematically because they differ only by constant scalar dimensional factors. One of these averaging was used in our papers [J. Garecki, Rep. Math. Phys. 33, 57 (1993); Int. J. Theor. Phys. 35, 2195 (1996); Rep. Math. Phys. 40, 485 (1997); J. Math. Phys. 40, 4035 (1999); Rep. Math. Phys. 43, 397 (1999); Rep. Math. Phys. 44, 95 (1999); Ann. Phys. (Leipzig) 11, 441 (2002); M.P. Dabrowski and J. Garecki, Class. Quantum. Grar. 19, 1 (2002)] giving the canonical superenergy and angular supermomentum tensors. In this paper we present another averaging of the differences of the energy–momentum and angular momentum which gives tensorial quantities with proper dimensions of the energy–momentum and angular momentum densities. We have called these tensorial quantities “the averaged relative energy–momentum and angular momentum tensors”. These tensors are very closely related to the canonical superenergy and angular supermomentum tensors and they depend on some fundamental length L > 0. The averaged relative energy–momentum and angular momentum tensors of the gravitational field obtained in the paper can be applied, like the canonical superenergy and angular supermomentum tensors, to coordinate independent analysis (local and in special cases also global) of this field. Up to now we have applied the averaged relative energy–momentum tensors to analyze vacuum gravitational energy and momentum and to analyze energy and momentum of the Friedman (and also more general, only homogeneous) universes. The obtained results are interesting, e.g., the averaged relative energy density is positive definite for the all Friedman and other universes which have been considered in this paper.      

The Extensions of Gravitational Soliton Solutions with Real Poles

We analyse vacuum gravitationalsoliton solutions with real poles in thecosmological context. It is well known that thesesolutions contain singularities on certain nullhypersurfaces. Using a Kasner seed solution, we demonstrate thatthese may contain thin sheets of null matter or may besimple coordinate singularities, and we describe anumber of possible extensions through them.     

Possible Resonator Configurations for the Spherical Gravitational Wave Antenna

We solve algebraically the equations of motion for a spherical antenna coupled to an arbitrary number of small resonators, free to move radially, and investigate the conditions under which damping forces can be neglected in the system. We show that in order that the antenna's modes be decoupled a preferred distribution of the resonators on its surface should be used. We find that either 5, 6, 10 or 16 resonators can be used as long as they are conveniently positioned on the antenna's surface. We calculate and analyse the frequency shift and the signal-to-noise ratio of the coupled system for the various distributions studied.     

Symmetric hyperbolic systems for a large class of fields in arbitrary dimension

Symmetric hyperbolic systems of equations are explicitly constructed for a general class of tensor fields by considering their structure as r-fold forms. The hyperbolizations depend on 2r−1 arbitrary timelike vectors. The importance of the so-called “superenergy” tensors, which provide the necessary symmetric positive matrices, is emphasized and made explicit. Thereby, a unified treatment of many physical systems is achieved, as well as of the sometimes called “higher order” systems. The characteristics of these symmetric hyperbolic systems are always physical, and directly related to the null directions of the superenergy tensor, which are in particular principal null directions of the tensor field solutions. Generic energy estimates and inequalities are presented too. Examples are included, in particular a mixed gravitational-scalar field system at the level of the Bianchi equations.     

Relativistic Hydrodynamic Cosmological Perturbations

Relativistic cosmological perturbation analysescan be made based on several different fundamental gaugeconditions. In the pressureless limit the variables incertain gauge conditions show the correct Newtonian behaviors. Considering the generalcurvature (K) and the cosmological constant ()in the background medium, the perturbed density in thecomoving gauge, and the perturbed velocity and the perturbed potential in the zero-shear gaugeshow the same behavior as the Newtonian ones in generalscales. In the first part, we elaborate these Newtoniancorrespondences. In the second part, using the identified gauge-in variant variables withcorrect Newtonian correspondences, we present therelativistic results with general pressures in thebackground and perturbation. We present the generalsuper-sound-horizon scale solutions of the above mentionedvariables valid for general K, Lambda, and generallyevolving equation of state. We show that, for vanishingK, the super-sound-horizon scale evolution ischaracterised by a conserved variable which is the perturbedthree-space curvature in the comoving gauge. We alsopresent equations for the multi-component hydrodynamicsituation and for the rotation and gravitational wave.     

A formulation of linearised gravity

The formulation of linearised gravity in terms of the electric and magnetic gravitational fields is extended to take into account the presence of matter. The modes of radiation, the equations of motion and the potential in the static case are given. The relevant components of the superenergy tensor are calculated and a quantity named the superforce is introduced.     

Time Multidimension in Gravity

The influence of possible additional (hidden) components of time on a body's motion in the field of a gravitational wave is considered. Contrary to the one-time theory, oscillations of the body height and width sizes in a plane perpendicular to the direction of the wave propagation occur independently from one another. This peculiarity can be used for the experimental check of emission of gravitational waves with distinct time trajectories in cosmic cataclysms. An interesting analogy between electromagnetic and gravitation quantities is discussed in the context of time multidimension.