A contrastive study on the influences of radial and three-dimensional satellite gravity gradiometry on the accuracy of the Earth's gravitational field recovery

The accuracy of the Earth’s gravitational field measured from the gravity field and steady-state ocean circulation explorer(GOCE),up to 250 degrees,influenced by the radial gravity gradient V zz and three-dimensional gravity gradient V ij from the satellite gravity gradiometry(SGG) are contrastively demonstrated based on the analytical error model and numerical simulation,respectively.Firstly,the new analytical error model of the cumulative geoid height,influenced by the radial gravity gradient V zz and three-dimensional gravity gradient V ij are established,respectively.In 250 degrees,the GOCE cumulative geoid height error measured by the radial gravity gradient V zz is about 2 1/2 times higher than that measured by the three-dimensional gravity gradient V ij.Secondly,the Earth’s gravitational field from GOCE completely up to 250 degrees is recovered using the radial gravity gradient V zz and three-dimensional gravity gradient V ij by numerical simulation,respectively.The study results show that when the measurement error of the gravity gradient is 3×10 12 /s 2,the cumulative geoid height errors using the radial gravity gradient V zz and three-dimensional gravity gradient V ij are 12.319 cm and 9.295 cm at 250 degrees,respectively.The accuracy of the cumulative geoid height using the three-dimensional gravity gradient V ij is improved by 30%-40% on average compared with that using the radial gravity gradient V zz in 250 degrees.Finally,by mutual verification of the analytical error model and numerical simulation,the orders of magnitude from the accuracies of the Earth’s gravitational field recovery make no substantial differences based on the radial and three-dimensional gravity gradients,respectively.Therefore,it is feasible to develop in advance a radial cold-atom interferometric gradiometer with a measurement accuracy of 10 13 /s 2-10 15 /s 2 for precisely producing the next-generation GOCE Follow-On Earth gravity field model with a high spatial resolution.     

Spherically symmetric solution in higher-dimensional teleparallel equivalent of general relativity

A theory of(N+1)-dimensional gravity is developed on the basis of the teleparallel equivalent of general relativity(TEGR).The fundamental gravitational field variables are the(N+1)-dimensional vector fields,defined globally on a manifold M,and the gravitational field is attributed to the torsion.The form of Lagrangian density is quadratic in torsion tensor.We then give an exact five-dimensional spherically symmetric solution(Schwarzschild(4+1)-dimensions).Finally,we calculate energy and spatial momentum using gravitational energy-momentum tensor and superpotential 2-form.     

Massive gravity as a quantum gauge theory

We present a new point of view on the quantization of the massive gravitational field, namely we use exclusively the quantum framework of the second quantization. The Hilbert space of the many-gravitons system is a Fock space F⁺ (H_graviton) where the one-particle Hilbert space H_graviton carries the direct sum of two unitary irreducible representations of the Poincaré group corresponding to two particles of mass m > 0 and spins 2 and 0, respectively. This Hilbert space is canonically isomorphic to a space of the type Ker(Q)/Im(Q) where Q is a gauge charge defined in an extension of the Hilbert space H_graviton generated by the gravitational field h and some ghosts fields u, (which are vector Fermi fields) and v (which is a vector Bose field).Then we study the self interaction of massive gravity in the causal framework. We obtain a solution which goes smoothly to the zero-mass solution of linear quantum gravity up to a term depending on the bosonic ghost field. This solution depends on two real constants as it should be; these constants are related to the gravitational constant and the cosmological constant. In the second order of the perturbation theory we do not need a Higgs field, in sharp contrast to Yang-Mills theory.     

A solution for galactic disks with Yukawian gravitational potential

We present a new solution for the rotation curves of galactic disks with gravitational potential of the Yukawa type. We follow the technique employed by Toomre in 1963 in the study of galactic disks in the Newtonian theory. This new solution allows an easy comparison between the Newtonian solution and the Yukawian one. Therefore, constraints on the parameters of theories of gravitation can be imposed, which in the weak field limit reduce to Yukawian potentials. We then apply our formulae to the study of rotation curves for a zero-thickness exponential disk and compare it with the Newtonian case studied by Freeman in 1970. As an application of the mathematical tool developed here, we show that in any theory of gravity with a massive graviton (this means a gravitational potential of the Yukawa type), a strong limit can be imposed on the mass (m _g) of this particle. For example, in order to obtain a galactic disk with a scale length of b∼ 10 kpc, we should have a massive graviton of m _g ≪ 10⁻⁵⁹g. This result is much more restrictive than those inferred from solar system observations.     

Renormalization of Quantum Field Theory in Curved Space-Time and Renormalization Group Equations

The structure of quantum field theory renormalization in curved space-time is investigated. The equations allowing us to investigate the behaviour of vacuum energy and vertex functions in the limit of small distances in the external gravitational field are established. The behaviour of effective charges corresponding to the parameters of nonminimal coupling of the matter with the gravitational field is studied and the conditions under which asymptotically free theories become asymptotically conformally invariant are found. The examples of asymptotically conformally invariant theories are given. On the basis of a direct solution of renormalization group equations the effective potential in the external gravitational field and the effective action in the gravity with the high derivatives are obtained. The expression for the cosmological constant in terms of R²-gravity Lagrangian parameters is given which does not contradict the observable data. Renormalization and renormalization group equations for the theory in curved space-time with torsion are investigated.     

Universal temperature corrections to the free energy for the gravitational field

The temperature correction to the free energy of the gravitational field is considered which does not depend on the Planck energy physics. The leading correction may be interpreted in terms of the temperature-dependent effective gravitational constant G_eff. The temperature correction to appears to be valid for all temperatures T?E_Planck. It is universal since it is determined only by the number of fermionic and bosonic fields with masses m?T, does not contain the Planck energy scale E_Planck which determines the gravitational constant at T=0, and does not depend on whether or not the gravitational field obeys the Einstein equations. That is why this universal modification of the free energy for gravitational field can be used to study thermodynamics of quantum systems in condensed matter (such as quantum liquids superfluid ³He and ⁴He), where the effective gravity emerging for fermionic and/or bosonic quasiparticles in the low-energy corner is quite different from the Einstein gravity.     

Spaces with flat symmetry in poincaré gauge theory of gravity

A Taub space is considered in the Poincare gauge theory of gravity. It is shown that the torsion tensor has four nonvanishing components, which can be split into two independent pairs S₀₁ ⁰, S₀₁ ¹, and S₂₃ ⁰, S₂₃ ¹. The analysis of the gravitational field equations leads to the conclusion that in this case only a flat space-time with torsion is possible, and that its metric coefficients and the components of the torsion tensor are described by a wave equation.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 4, pp. 92–98, April, 1990.     

Thermal undulations of quasi-spherical vesicles stabilized by gravity

The classical treatment of quasi-spherical vesicle undulations has, in the present work, been reviewed and extended to systems, which are affected by a gravitational field caused by a density difference across the membrane. The effects have been studied by the use of perturbation theory leading to corrections to the mean shape and the fluctuation correlation matrix. These corrections have been included in an analytical expression for the flicker spectrum to probe how the experimentally accessible spectrum changes with gravity. The results are represented in terms of the gravitational parameter, g ₀ = ΔρgR ⁴/κ. The contributions from gravity are in most experimental situations small and thus negligible, but for values of g₀ above a certain limit, the perturbational corrections must be included. Expressions for the relative error on the flicker spectrum have been worked out, so that it is possible to define the regime where gravity is negligible. An upper limit of g₀ has also been identified, where the error in all modes of the flicker spectrum is significant due to distortion of the mean shape. Received 9 July 2002 and Received in final form 15 November 2002 RID="a" ID="a"e-mail: jonas@kemi.dtu.dk RID="b" ID="b"e-mail: ipsen@memphys.sdu.dk     

Quantum Non-Gravity and Stellar Collapse

Observational indications combined with analyses of analogue and emergent gravity in condensed matter systems support the possibility that there might be two distinct energy scales related to quantum gravity: the scale that sets the onset of quantum gravitational effects E_BE_{\rm B} (related to the Planck scale) and the much higher scale E_LE_{\rm L} signalling the breaking of Lorentz symmetry. We suggest a natural interpretation for these two scales: E_LE_{\rm L} is the energy scale below which a special relativistic spacetime emerges, E_BE_{\rm B} is the scale below which this spacetime geometry becomes curved. This implies that the first ‘quantum’ gravitational effect around E_BE_{\rm B} could simply be that gravity is progressively switched off, leaving an effective Minkowski quantum field theory up to much higher energies of the order of E_LE_{\rm L}. This scenario may have important consequences for gravitational collapse, inasmuch as it opens up new possibilities for the final state of stellar collapse other than an evaporating black hole.     

On the Theory of Gravitational Field in Finsler Spaces

Some structural considerations are made on the Finslerian gravitational field: A Finslerian metrical structure such as g_λχ(x, y) = γ_λχ(x) + h_λχ(x, y) is proposed, where γ_λχ denotes the Riemann metric of Einstein's gravitational field, while hλχ the Finsler metric induced by the Riemann metric h_ij(y) of the internal field; The intrinsic behaviour of the internal variable y, which is expressed as ?ⁱ = K(x, y) y^j in the internal field, is grasped by the Finslerian parallelism δyⁱ (=0), which is reflected in the spatial structure of the external gravitational field by the mapping relation δy^χ = e(x) δyⁱ. The whole metrical Finsler connection D for g_λχ(i.e., Dg_λχ = 0) is determined by taking account of the intrinsic behaviour δy^χ.     

General relativity in a (2+1)-dimensional space-time: An electrically charged solution

We have found a static electrically charged solution to the Einstein-Maxwell equations in a (2+1)-dimensional space-time. Studies of general relativity in lower dimensional space-times provide many new insights and a simplified arena for doing quantum mechanics. In (2+1)-dimensional space-time, solutions to the vacuum field equations are locally flat (point masses are conical sigularities), but when electromagnetic fields are presentT ^ab O and the solutions are curved. For a static chargeQ we find andds ²= –(kQ ² /2)In(r _c /r)dt ² + (2/kQ ²[ln(r _c /r)]^–1 dr ² +r ² d ² wherer _c is a constant. There is a horizon atr =r _c like the inner horizon of the Reisner-Nordström solution. We have produced a Kruskal extension of this metric which shows two static regions (I and III) withr <r _c and two dynamical regions (II and IV) withr>r _c . A spacelike slice across regions I and III shows a football-shaped universe with chargeQ at one end and –Q at the other. Slices in the dynamical regions (II and IV) show a cylindrical universe that is expanding in region II and contracting in region IV. Electromagnetic solutions to the Einstein-Maxwell field equations in lower dimensional space-times can be used to provide new insights into Kaluza-Klein theories. In terms of the Kaluza-Klein theory, for example, electromagnetic radiation in a (2+1)-dimensional space-time is really gravitational radiation in the associated (3+1)-dimensional Kaluza-Klein space-time. According to Kaluza Klein theory the absence of gravitational radiation in (2+1)-dimensional space-time implies (correctly) the absence of electromagnetic radiation in (1+1)-dimensional space-time.     

Repulsive gravity induced by a conformally coupled scalar field implies a bouncing radiation-dominated universe

In the present work we revisit a model consisting of a scalar field with a quartic self-interaction potential non-minimally (conformally) coupled to gravity (Novello in Phys Lett 90A:347 1980). When the scalar field vacuum is in a broken symmetry state, an effective gravitational constant emerges which, in certain regimes, can lead to gravitational repulsive effects when only ordinary radiation is coupled to gravity. In this case, a bouncing universe is shown to be the only cosmological solution admissible by the field equations when the scalar field is in such broken symmetry state.     

Mass and dual mass: A gravitational analogue of the Dirac quantization rule and its physical implications

It is shown that (asymptotically multi-NUT) gravitational magnetic monopoles, which can be described by anS ³/Z _N principal Hopf-bundle structure at conformal null infinity (Z _N is a cyclic subgroup of orderN ofZ). provide a gravitational analogue of the Dirac quantization rule, which involves the total magnetic (dual) mass of the space-time-a measurement of the first Chern class of the bundle-and the mass of a test particle located in the rest frame defined at infinity by the Bondi (or dual Bondi) 4-momentum. It is shown thatSU ₂/U ₁ preserves the asymptotic structure. A definition of the angular momentum operator which extends that available for test electric charges in the field of a (Maxwellian) Yang-Wu magnetic monopole is presented. The commutation relations are dictated by the quantization rule. Various physical consequences are mentioned. SinceSU ₂ is a double covering ofSO ₃, gravitational magnetic monopoles provide a topological explanation for the existence of particles with half-integer spin. Abelian (U ₁), non-AbelianSU ₁ asymptotic degrees of freedom of the gravitational field could be related to suitable nontrivial cohomology classes; Penrose's nonlinear graviton modes could emerge as self (antiself) adjoint (Yang-Mills) gauge connections.     

Scale Transformation,Modified Gravity,and Brans-Dicke Theory

A model of Einstein-Hilbert action subject to the scale transformation is studied. By introducing a dilaton field as a means of scale transformation a new action is obtained whose Einstein field equations are consistent with traceless matter with non-vanishing modified terms together with dynamical cosmological and gravitational coupling terms. The obtained modified Einstein equations are neither those in f(R) metric formalism nor the ones in f(ℛ) Palatini formalism, whereas the modified source terms are formally equivalent to those of f(R)=\frac12R²f({\mathcal{R}})=\frac{1}{2}{\mathcal{R}}^{2} gravity in Palatini formalism. The correspondence between the present model, the modified gravity theory, and Brans-Dicke theory with w = -\frac32\omega=-\frac{3}{2} is explicitly shown, provided the dilaton field is condensated to its vacuum state.     

Confinement of tensor gluons — a classical approach

We look at the confinement of tensor gluons (f _μν ^(c) field) in a strong gravity background and find that the strong gravity provides a trap for the confinement of colour waves of selected frequencies. We assume that the tensorf _μν ^(c) field (mediating quanta: tensor 2⁺ f-meson) satisfies Einstein-like equations with a cosmological constant. The colour field satisfy equations resembling Maxwell form of the linear theory of gravitation and see the effect off _μν ^(c) field as playing the role of a medium having space dependent dielectric permeabilities. The solution of colour field equations resemble harmonic oscillator type wave functions with equispaced energy levels (no continuum) leading to confinement.     

The linear theory of gravitation in the radiation gauge

A global gauge for the linear theory of gravity is given which avoids the (In(R))/R problem of harmonic coordinates. Consequently it is possible to do 1/R expansions for precisely those null sources which lead to difficulties in the harmonic gauge. This is important, for example, in second order calculations where the first order field is a gravitational wave. This gauge makes manifest the two degrees of freedom for the dynamic fields. In it the second time derivative of the metric itself has physical significance since R _oioj=–1/2h_ij.Work supported in part by NSF Grant No. GP-13959.     

The preferred null symmetries of a null electromagnetic field interacting with a null free gravitational field

The real null vector 1 ^a of the Newman-Penrose formalism is preferred to correspond to a geometrical symmetry as well as a dynamical symmetry. The 16 types of geometrical symmetries expressed through the vanishing of the Lie derivatives of certain tensor fields with respect to 1 ^a are examined separately. Two types of dynamical symmetries are imposed simultaneously on 1 ^a : A null electromagnetic field and a null gravitational field are both chosen to have the same propagation vector 1 ^a . By adopting freedom conditions on 1 ^a , it is shown that the symmetries of the null electromagnetic field are shared neither by the free gravitational field nor by the gravitational potentials. In fact the following five preferred null symmmetries are found to be proper: motion, affine collineation, special curvature collineation, curvature collineation, and Ricci collineation. The scalars characterizing the coupled fields are found to be constant with respect to 1 ^a .     

Charged axially symmetric solution and energy in teleparallel theory equivalent to general relativity

An exact charged solution with axial symmetry is obtained in the teleparallel equivalent of general relativity. The associated metric has the structure function G(ξ)=1-ξ²-2mAξ³-q²A²ξ⁴. The fourth order nature of the structure function can make calculations cumbersome. Using a coordinate transformation we get a tetrad whose metric has the structure function in a factorizable form (1-ξ²)(1+r₊Aξ)(1+r_-Aξ) with r_± as the horizons of Reissner–Nordström space-time. This new form has the advantage that its roots are now trivial to write down. Then, we study the singularities of this space-time. Using another coordinate transformation, we obtain a tetrad field. Its associated metric yields the Reissner–Nordström black hole. In calculating the energy content of this tetrad field using the gravitational energy-momentum, we find that the resulting form depends on the radial coordinate! Using the regularized expression of the gravitational energy-momentum in the teleparallel equivalent of general relativity we get a consistent value for the energy.     

General models of Einstein gravity with a non-Newtonian weak-field limit

We investigate Einstein theories of gravity, coupled to a scalar field j{\varphi} and point-like matter, which are characterized by a scalar field-dependent matter coupling function e^H(j){e^{H(\varphi)}} . We show that under mild constraints on the form of the potential for the scalar field, there are a broad class of Einstein-like gravity models—characterized by the asymptotic behavior of H—which allow for a non-Newtonian weak-field limit with the gravitational potential behaving for large distances as ln r. The Newtonian term GM/r appears only as sub-leading. We point out that this behavior is also shared by gravity models described by f (R) Lagrangians. The relevance of our results for the building of infrared modified theories of gravity and for modified Newtonian dynamics is also discussed.     

The validity of the formulaE=mc 2 for gravitational energy

Theoretical evidence for the validity of the formulaE =mc ² for gravitational energy in general relativity is reviewed. For isolated bodies the formula applies to both the inertial mass and the mass as a source of gravity. The formula also holds for the case of the mass density (as a source of gravity) of an inhomogeneous medium with small-scale gravitational interactions.