Gravitational Term Research Articles

THE NATURE OF THE COSMOLOGICAL CONSTANT PROBLEM

General relativity postulates the Minkowski space-time as the standard (flat) geometry against which we compare all curved space-times and also as the gravitational ground state where particles, quantum fields and their vacua are defined. On the other hand, experimental evidences tell that there exists a non-zero cosmological constant, which implies in a deSitter ground state, which not compatible with the assumed Minkowski structure. Such inconsistency is an evidence of the missing standard of curvature in Riemann's geometry, which in general relativity manifests itself in the form of the cosmological constant problem. We show how the lack of a curvature standard in Riemann's geometry can be fixed by Nash's theorem on metric perturbations. The resulting higher dimensional gravitational theory is more general than general relativity, similar to brane-world gravity, but where the propagation of the gravitational field along the extra dimensions is a mathematical necessity, rather than a postulate. After a brief introduction to Nash's theorem, we show that the vacuum energy density must remain confined to four-dimensional space-times, but the cosmological constant resulting from the contracted Bianchi identity represents a gravitational term which is not confined. In this case, the comparison between the vacuum energy and the cosmological constant in general relativity does not make sense. Instead, the geometrical fix provided by Nash's theorem suggests that the vacuum energy density contributes to the perturbations of the gravitational field.

DOES THE HETEROTIC SUPERSTRING THEORY CONTAIN CLOSED TIME-LIKE LINES?

Causal solutions of the Gödel type, for which the line element is ds2 = dt2 - 2b e mxdtdv - c e 2mxdv2 - dx2 - dz2 with c = 0, are known to exist for gravitational theories containing a cosmological constant Λ and quadratic higher-derivative terms defined by the Lagrangian L = -(1/2)κ-2(R + 2Λ) + A1R2 + A2RijRij. Here, we show that acausal solutions, for which c < 0, containing closed time-like lines, can be constructed only if A2 = 0. Extension of this analysis to the heterotic superstring theory, including a generic massless scalar field ϕ plus quadratic and quartic gravitational terms [Formula: see text] and [Formula: see text], again yields a causal solution with c = 0, and also Λ = 0, as required for anomaly freedom, while solutions with c < 0 are ruled out. More general rotational space–times appear to be intractable analytically, and therefore it remains a matter of conjecture that the heterotic superstring admits only classical Lorentzian solutions which respect causality. For the energy density ρ(ϕ) of the scalar field is positive semi-definite only when g00 ≥ 0, which is equivalent to the causality condition g11 ≤ 0 or c ≥ 0 in a Lorentzian space–time for which det gij < 0; while ρ(ϕ) is unbounded from below in the presence of closed time-like lines, when g11 > 0, implying instability of ϕ, which will react back on the metric until it becomes causal.

On the gravitational content of molecular clouds and their cores

The gravitational term for clouds and cores entering in the virial theorem is usually assumed to be equal to the gravitational energy, since the contribution to the gravitational force from the mass distribution outside the volume of integration is assumed to be negligible. Such approximation may not be valid in the presence of an important external net potential. In the present work, we analyse the effect of an external gravitational field on the gravitational budget of a density structure. Our cases under analysis are (a) a giant molecular cloud (GMC) with different aspect ratios embedded within a galactic net potential, including the effects of gravity, shear and inertial forces, and (b) a molecular cloud (MC) core embedded within the gravitational potential of its parent MC. We find that for roundish GMCs, the tidal tearing due to the shear in the plane of the Galaxy is compensated by the tidal compression in the z-direction. The influence of the external effective potential on the total gravitational budget of these clouds is relatively small (up to ∼15–25 per cent), although not necessarily negligible. However, for more filamentary GMCs, elongated on the plane of the Galaxy, the external effective potential can be dominant and can even overwhelm self-gravity, regardless of whether its main effect on the cloud is to disrupt it or compress it. This may explain the presence of some GMCs with few or no signs of massive star formation, such as the Taurus or the Maddalena’s clouds. In the case of dense cores embedded in their parent MC, we found that the gravitational content due to the external field may be more important than the gravitational energy of the cores themselves. This effect works in the same direction as the gravitational energy, that is favouring the collapse of cores. We speculate on the implications of these results for star formation models, in particular that apparently nearly magnetically critical cores may actually be supercritical due to the effect of the external potential.

On the heterotic string loop expansion

The effective action for the bosonic sector of the heterotic superstring theory, considered as an expansion in string loops, is reduced to four dimensions, with particular regard to the effects of the one-loop, parity-even term involving the eight-form X 8 introduced by Green et al. Formulae are given for the one-loop corrections to the ten-dimensional, tree-level quartic gravitational term R ˆ 4 → R ˆ ′ 4 , which determines the renormalization of the four-dimensional quadratic term R 2 via ( R ˆ 4 → R 2 R ¯ 2 ) → ( R ˆ ′ 4 → R ′ 2 R ¯ 2 ) . We also derive the formula for the four-dimensional, gauge kinetic function up to two loops, f = A r + β B r + γ A r −1 B r 4 , where A r and B r are moduli and γ is the unknown two-loop coefficient of the E 8 × E 8 ′ string. In the theory of gaugino condensation in the hidden sector, based upon the no-scale supergravity theory, the parameter β determines the ratio of the masses of the gaugino and gravitino to one loop, ξ ≡ m 1 / 2 / m 3 / 2 ≈ β B r / ( A r + β B r ) . The torus approximation for the internal space g ¯ μ ν predicts, setting A r ≈ 2 , B r ≈ 3 , that this ratio is close to unity, ξ ≈ 12 / 13 , while the six-sphere would yield a somewhat smaller ratio, ξ ≈ 1 / 15 .

Dynamical couplings reduction for rigid manipulators using generalized velocity components

A method for evaluation and reduction of dynamical couplings between the links of a rigid serial manipulator is presented. The approach is based on the first-order equations of motion expressed in terms of the generalized velocity components (GVC). The GVC obtained from decomposition of the manipulator mass matrix reflect the dynamics of the system. In order to reduce the dynamical couplings several criteria and indexes are introduced. As a result of the reduction, the time response of the system with a controller is improved. Moreover, the joint applied torques, the Coriolis and the gravitational term forces, and the kinetic energy are reduced too. The proposed method is tested via simulation on a spatial manipulator with three degrees of freedom.

THE AVERAGING PROBLEM IN COSMOLOGY AND MACROSCOPIC GRAVITY

The averaging problem in cosmology and the approach of macroscopic gravity to resolve the problem is discussed. The averaged Einstein equations of macroscopic gravity are modified on cosmological scales by the macroscopic gravitational correlation tensor terms as compared with the Einstein equations of general relativity. This correlation tensor satisfies a system of structure and field equations. An exact cosmological solution to the macroscopic gravity equations for a constant macroscopic gravitational connection correlation tensor for a flat spatially homogeneous, isotropic macroscopic space-time is presented. The correlation tensor term in the macroscopic Einstein equations has been found to take the form of either a negative or positive spatial curvature term. Thus, macroscopic gravity provides a cosmological model for a flat spatially homogeneous, isotropic Universe which obeys the dynamical law for either an open or closed Universe.

GOLDSTONE FIELDS IN THE SUPERSTRING THEORY

If global supersymmetry is broken by gaugino condensation in the hidden sector of the [Formula: see text] heterotic superstring theory after compactification, then the auxiliary field FBof the modulus B ≡ (Br, Bi) attains a finite value, while that of the dilaton A ≡ (Ar, Ai) vanishes, FA= 0, the Goldstone fermion being the modulino [Formula: see text], the spin-½ component of the complex chiral supermultiplet [Formula: see text]. The Goldstone boson of the scale symmetry that is broken when the radius of the internal space is fixed at a constant value is Br, which is determined from the goldstino Lagrangian, compared term by term with the superstring Lagrangian, including higher-derivative gravitational terms [Formula: see text] and [Formula: see text], after linking the space–time curvature to the energy–momentum tensor of the goldstino via the Einstein equations. This non-linear formulation of supersymmetry, due to Volkov and Akulov, is expressed in terms of the goldstino alone, whose Lagrangian contains a negative cosmological constant, which can be cancelled by the super-Higgs mechanism of Deser and Zumino to make the gravitino massive and break supersymmetry at the level [Formula: see text] GeV, while [Formula: see text]. Here, the modulus has been scaled to the Hagedorn value for the heterotic superstring theory, [Formula: see text], and Ar, identified with the inverse square of the tree-level gauge coupling, has been scaled to the calculation in the minimal supersymmetric standard model due to Weinberg, that g-2= 1.39 at the unification mass MX= 2.2 × 1016GeV, assuming three generations of elementary particles and two Higgs doublets. In the presence of gravitino condensation in the internal space, however, there is an arbitrary additional contribution to the cosmological constant, facilitating reduction of msto ~ 100 TeV, say, and m3/2to ~ 1 eV.

Equations of motion and ballistic paths of volcanic ejecta

When the equations of motion are broken down into component form their solution provides the coordinates of positions of moving particles at successive times, making it possible to plot the paths of those particles. In two dimensions the equations contain horizontal and vertical accelerations which, when added separately in each component's direction, provide the net force per unit mass acting horizontally and vertically. Repeated integration of accelerations leads to particle velocities and displacements, hence particle positions and paths. Though conceptually applicable to many kinds of motion in geomorphology, the method is applied here to paths of volcanic ejecta in equatorial latitudes. The horizontal component includes the sum of inertial, resistance and aeolian terms whereas the vertical component is the sum of inertial, gravitational, resistance, lift and total centrifugal terms. The total centrifugal effect on a body in motion on a rotating Earth is itself composed of two components. One component is the conventionally conceived centrifugal effect at right angles to Earth's polar axis. The second component is the Coriolis effect at right angles to the velocity vectors that are tangential to the path of motion and at right angles to the axis of spin of the moving body. When the axis of spin is coincident with the polar axis, as it is for eastward motion of a body on the equator, both components act in the same direction, in this case upwards vertically. Completing the scenario is the inclusion of the equatorial tropopause which impedes further vertical motion but allows horizontal motion to continue, an aspect of considerable importance with respect to transport of dust (volcanic and aeolian) and pollutants.

Underthrusting‐accretion cycle: Work budget as revealed by the boundary element method

Sandbox models of accretionary wedges have demonstrated that fault systems grow episodically via cycles of alternating wedge thickening, which is accommodated by slip along faults within the wedge (underthrusting), and wedge lengthening, which is accommodated by growth of new faults at the wedge toe (accretion). The transition between these two modes of deformation is controlled by the interplay of work against gravity, frictional heating, the work of deformation around faults, and the work of fault propagation and seismic/acoustic energy. Using numerical mechanical models based on the boundary element method, we have simulated the deformation observed in sandbox experiments, providing a mechanical analysis of the underthrusting/accretion transition. Our results show that the total work done by the contracting wedge increases during the underthrusting stage up to a critical value when the propagation of a new frontal thrust significantly reduces the work required for further deformation. The numerical models also predict the location of the maximum shear along the basal décollement during underthrusting as well as the energetically most viable position and vergence for the nucleation of a new thrust. These locations do not coincide, and the match of the energetically most favorable position with the experimental results suggests that the new thrust ramps develop first ahead and then link down and backward to the propagating basal décollement. The shear localization producing a new thrust ramp will occur where the energy spent by the deforming wedge is minimized due to an optimal combination of gravitational, frictional, internal, and propagation work terms.

Nonadiabatic charged spherical gravitational collapse

We present a complete set of the equations and matching conditions required for the description of physically meaningful charged, dissipative, spherically symmetric gravitational collapse with shear. Dissipation is described with both free-streaming and diffusion approximations. The effects of viscosity are also taken into account. The roles of different terms in the dynamical equation are analyzed in detail. The dynamical equation is coupled to a causal transport equation in the context of Israel-Stewart theory. The decrease of the inertial mass density of the fluid, by a factor which depends on its internal thermodynamic state, is reobtained, with the viscosity terms included. In accordance with the equivalence principle, the same decrease factor is obtained for the gravitational force term. The effect of the electric charge on the relation between the Weyl tensor and the inhomogeneity of the energy density is discussed.

Irreducible forms for the metric variations of the action terms of sixth-order gravity and approximated stress–energy tensor

We provide irreducible expressions for the metric variations of the gravitational action terms constructed from the 17 curvature invariants of order six in the derivatives of the metric tensor, i.e., from the geometrical terms appearing in the diagonal heat-kernel or Gilkey–DeWitt coefficient a3. We then express, for a four-dimensional spacetime, the approximated stress–energy tensor constructed from the renormalized DeWitt–Schwinger effective action associated with a massive scalar field. We also construct, for higher dimensional spacetimes, the infinite counterterms of order six in the derivatives of the metric tensor appearing on the left-hand side of Einstein equations as well as the contribution associated with the cubic Lovelock gravitational action. In the appendix, we provide a list of geometrical relations we have used and which are more generally helpful for calculations in two-loop quantum gravity in a four-dimensional background or for calculations in one-loop quantum gravity in higher dimensional background. We also obtain the approximated stress–energy tensors associated with a massive spinor field and a massive vector field propagating in a four-dimensional background.

Quantum Gravity Boundary Terms from the Spectral Action of Noncommutative Space

We study the boundary terms of the spectral action of the noncommutative space, defined by the spectral triple dictated by the physical spectrum of the standard model, unifying gravity with all other fundamental interactions. We prove that the spectral action predicts uniquely the gravitational boundary term required for consistency of quantum gravity with the correct sign and coefficient. This is a remarkable result given the lack of freedom in the spectral action to tune this term.

A three-dimensional multidimensional gas-kinetic scheme for the Navier–Stokes equations under gravitational fields

This paper extends the gas-kinetic scheme for one-dimensional inviscid shallow water equations (K. Xu, A well-balanced gas-kinetic scheme for the shallow-water equations with source terms, J. Comput. Phys. 178 (2002) 533–562) to multidimensional gas dynamic equations under gravitational fields. Four important issues in the construction of a well-balanced scheme for gas dynamic equations are addressed. First, the inclusion of the gravitational source term into the flux function is necessary. Second, to achieve second-order accuracy of a well-balanced scheme, the Chapman–Enskog expansion of the Boltzmann equation with the inclusion of the external force term is used. Third, to avoid artificial heating in an isolated system under a gravitational field, the source term treatment inside each cell has to be evaluated consistently with the flux evaluation at the cell interface. Fourth, the multidimensional approach with the inclusion of tangential gradients in two-dimensional and three-dimensional cases becomes important in order to maintain the accuracy of the scheme. Many numerical examples are used to validate the above issues, which include the comparison between the solutions from the current scheme and the Strang splitting method. The methodology developed in this paper can also be applied to other systems, such as semi-conductor device simulations under electric fields.

Equilibrium and stability of interstellar medium in its own gravitational field

The statistical equilibrium of a heterogeneous self-gravitating layer of the interstellar medium is examined. Electrostatic fields and the different reactions to them by electrons and the ions of different elements are taken into account. In general, different, uncoupled temperatures are specified for the individual components of the interstellar medium. It is emphasized that some details of the structure of an equilibrium layer cannot be interpreted in terms of gravitation alone, despite the weakness of the electrostatic fields. A mixture of H II, He II, and electrons is examined as an important example. The asymptotic behavior of the total density and of the ratio of partial densities over large distances is found. A method for finding (in quadrature) these characteristics at any point of the medium is described.

Centrifugal effects in rotating convection: axisymmetric states and three-dimensional instabilities

Rotating convection is analysed numerically in a cylinder of aspect ratio one, for Prandtl number about 7. Traditionally, the problem has been studied within the Boussinesq approximation with density variation only incorporated in the gravitational buoyancy term and not in the centrifugal buoyancy term. In that limit, the governing equations admit a trivial conduction solution. However, the centrifugal buoyancy changes the problem in a fundamental manner, driving a large-scale circulation in which cool denser fluid is centrifuged radially outward and warm less-dense fluid is centrifuged radially inward, and so there is no trivial conduction state. For small Froude numbers, the transition to three-dimensional flow occurs for Rayleigh number R ≈ 7.5 × 103. For Froude numbers larger than 0.4, the centrifugal buoyancy stabilizes the axisymmetric large-scale circulation flow in the parameter range explored (up to R = 3.5 × 104). At intermediate Froude numbers, the transition to three-dimensional flow is via four different Hopf bifurcations, resulting in different coexisting branches of three-dimensional solutions. How the centrifugal and the gravitational buoyancies interact and compete, and the manner in which the flow becomes three-dimensional is different along each branch. The centrifugal buoyancy, even for relatively small Froude numbers, leads to quantitative and qualitative changes in the flow dynamics.

METRIC-SCALAR GRAVITY WITH TORSION AND THE MEASURABILITY OF THE NON-MINIMAL COUPLING

The "measurability" of the non-minimal coupling is discussed by considering the correction to the Newtonian static potential in the semiclassical approach. The coefficient of the "gravitational Darwin term" (GDT) gets redefined by the non-minimal torsion scalar couplings. Based on a similar analysis of the GDT in the effective field theory approach to non-minimal scalar, we conclude that for reasonable values of the couplings the correction is very small.

Averaging in spherically symmetric cosmology

The averaging problem in cosmology is of fundamental importance. When applied to study cosmological evolution, the theory of macroscopic gravity (MG) can be regarded as a long-distance modification of general relativity. In the MG approach to the averaging problem in cosmology, the Einstein field equations on cosmological scales are modified by appropriate gravitational correlation terms. We study the averaging problem within the class of spherically symmetric cosmological models. That is, we shall take the microscopic equations and effect the averaging procedure to determine the precise form of the correlation tensor in this case. In particular, by working in volume-preserving coordinates, we calculate the form of the correlation tensor under some reasonable assumptions on the form for the inhomogeneous gravitational field and matter distribution. We find that the correlation tensor in a Friedmann-Lema\^{\i}tre-Robertson-Walker (FLRW) background must be of the form of a spatial curvature. Inhomogeneities and spatial averaging, through this spatial curvature correction term, can have a very significant dynamical effect on the dynamics of the Universe and cosmological observations; in particular, we discuss whether spatial averaging might lead to a more conservative explanation of the observed acceleration of the Universe (without the introduction of exotic dark matter fields). We also find that the correlation tensor for a non-FLRW background can be interpreted as the sum of a spatial curvature and an anisotropic fluid. This may lead to interesting effects of averaging on astrophysical scales. We also discuss the results of averaging an inhomogeneous Lema\^{\i}tre-Tolman-Bondi solution as well as calculations of linear perturbations (that is, the backreaction) in an FLRW background, which support the main conclusions of the analysis.

The Virial Balance of Clumps and Cores in Molecular Clouds

We study the instantaneous virial balance of clumps and cores (CCs) in 3D simulations of driven, MHD, isothermal molecular clouds (MCs). The models represent a range of magnetic field strengths in MCs from subcritical to non-magnetic regimes. We identify CCs at different density thresholds, and for each object, we calculate all the terms that enter the Eulerian form of the virial theorem (EVT). A CC is considered gravitationally bound when the gravitational term in the EVT is larger than the amount for the system to be virialized, which is more stringent than the condition that it be large enough to make the total volume energy negative. We also calculate, quantities commonly used in the observations to indicate the state of gravitational boundedness of CCs such as the Jeans number J_c, the mass-to magnetic flux ratio mu_c, and the virial parameter alpha_vir. Our results show that: a) CCs are dynamical out-of-equilibrium structures. b) The surface energies are of the same order than their volume counterparts c) CCs are either in the process of being compressed or dispersed by the velocity field. Yet, not all CCs that have a compressive net kinetic energy are gravitationally bound. d) There is no 1-to-1 correspondence between the state of gravitational boundedness of a CC as described by the virial analysis or as implied by the classical indicators. In general, in the virial analysis, we observe that only the inner regions of the objects are gravitationally bound, whereas J_c, alpha_vir, and mu_c estimates tend to show that they are more gravitationally bound at the lowest threshold levels and more magnetically supercritical. g) We observe, in the non-magnetic simulation, the existence of a bound core with structural and dynamical properties that resemble those of the Bok globule Barnard 68 (B68).

INTRODUCTION TO MODIFIED GRAVITY AND GRAVITATIONAL ALTERNATIVE FOR DARK ENERGY

We review various modified gravities considered as gravitational alternative for dark energy. Specifically, we consider the versions of f(R), f(G) or f(R, G) gravity, model with non-linear gravitational coupling or string-inspired model with Gauss-Bonnet-dilaton coupling in the late universe where they lead to cosmic speed-up. It is shown that some of such theories may pass the Solar System tests. On the same time, it is demonstrated that they have quite rich cosmological structure: they may naturally describe the effective (cosmological constant, quintessence or phantom) late-time era with a possible transition from decceleration to acceleration thanks to gravitational terms which increase with scalar curvature decrease. The possibility to explain the coincidence problem as the manifestation of the universe expansion in such models is mentioned. The late (phantom or quintessence) universe filled with dark fluid with inhomogeneous equation of state (where inhomogeneous terms are originated from the modified gravity) is also described.

Global solution to the one-dimensional equations for a self-gravitating viscous radiative and reactive gas

In this paper we consider a system of equations describing a motion of a self-gravitating one-dimensional gaseous medium in the presence of radiation and reacting process. By introducing Lagrangian mass coordinate, this free-boundary problem is reduced to the problem in a fixed domain with an explicit gravitational term. Based on the fundamental local existence result and a priori estimates, we can construct a classical unique global solution.