Simple Derivation of Minimum Length, Minimum Dipole Moment and Lack of Space–Time Continuity

The principle of maximum power makes it possible to summarize special relativity, quantum theory and general relativity in one fundamental limit principle each. Special relativity contains an upper limit to speed; following Bohr, quantum theory is based on a lower limit to action; recently, a maximum power given by c ⁵/4G was shown to be equivalent to the full field equations of general relativity. Taken together, these three fundamental principles imply a limit value for every physical observable, from acceleration to size. The new, precise limit values differ from the usual Planck values by numerical prefactors of order unity. Among others, minimum length and time intervals appear. The limits imply that elementary particles are not point-like and suggest a lower limit on electric dipole values. The minimum intervals also imply that the non-continuity of space–time is an inevitable result of the unification of quantum theory and relativity, independently of the approach used. PACS numbers: 04.20.Cv; 13.40.Em; 04.60.-m.     

Unification of electromagnetism and gravitation: The equations of electrodynamics derived from the Riemann tensor in general relativity

The equations of free-space electrodynamics are derived directly from the Riemann curvature tensor and the Bianchi identity of general relativity by contracting on two indices to give a novel antisymmetric Ricci tensor. Within a factore/h, this is the field-strength tensor G of free-space electrodynamics. The Bianchi identity for G describes free-space electrodynamics in a manner analogous to, but more general than, Maxwell's equations for electrodynamics, the critical difference being the existence in general and special relativity of the Evans-Vigier fieldB ⁽³⁾.     

Gödel type metrics in Einstein–aether theory

Aether theory is introduced to implement the violation of the Lorentz invariance in general relativity. For this purpose a unit timelike vector field is introduced to the theory in addition to the metric tensor. Aether theory contains four free parameters which satisfy some inequalities in order that the theory to be consistent with the observations. We show that the Gödel type of metrics of general relativity are also exact solutions of the Einstein–aether theory. The only field equations are the 3D Maxwell field equations and the parameters are left free except c ₁ ? c ₃ = 1.     

Bimetric general relativity and cosmology

A modification of the general relativity theory is proposed (bimetric general relativity) in which, in addition to the usual metric tensorg _v describing the space-time geometry and gravitation, there exists also a background metric tensor _v The latter describes the space-time of the universe if no matter were present and is taken to correspond to a space-time of constant curvature with positive spatial curvature (k=1). Field equations are obtained, and these agree with the Einstein equations for systems that are small compared to the size of the universe, such as the solar system. Energy considerations lead to a generalized form of the De Donder condition. One can set up simple isotropic closed models of the universe which first contract and then expand without going through a singular state. It is suggested that the maximum density of the universe was of the order ofc ^{5 –1} G ^–210⁹³ g/cm³. The expansion from such a high-density state is similar to that from the singular state (big bang) of the general relativity models. In the case of the dust-filled model one can fit the parameters to present cosmological data. Using the radiation-filled model to describe the early history of the universe, one can account for the cosmic abundance of helium and other light elements in the same way as in ordinary general relativity.     

Modified <Emphasis Type="Italic">f</Emphasis>(<Emphasis Type="Italic">G</Emphasis>) gravity models with curvature–matter coupling

A modified f(G) gravity model with coupling between matter and geometry is proposed, which is described by the product of the Lagrange density of the matter and an arbitrary function of the Gauss–Bonnet term. The field equations and the equations of motion corresponding to this model show the non-conservation of the energy-momentum tensor, the presence of an extra force acting on test particles and non-geodesic motion. Moreover, the energy conditions and the stability criterion at the de Sitter point in modified f(G) gravity models with curvature–matter coupling are derived, which can degenerate to the well-known energy conditions in general relativity. Furthermore, in order to get some insight in the meaning of these energy conditions, we apply them to the specific models of f(G) gravity and the corresponding constraints on the models are given. In addition, the conditions and the candidate for late-time cosmic accelerated expansion in modified f(G) gravity are studied by means of conditions of power-law expansion and the equation of state of matter ω smaller than -\frac13-\frac{1}{3}.     

Teleparallel equivalent theory of (1+1)-dimensional gravity

A theory of (1+1)-dimensional gravity is constructed on the basis of the teleparallel equivalent of general relativity.The fundamental field variables are the tetrad fields e i μ and the gravity is attributed to the torsion.A dilatonic spherically symmetric exact solution of the gravitational field equations characterized by two parameters M and Q is derived.The energy associated with this solution is calculated using the two-dimensional gravitational energy-momentum formula.     

A class of wormhole solutions to higher-dimensional general relativity

An ansatz is given which reduces the equations of sourceless (n+p)-dimensional general relativity to those ofn-dimensional general relativity coupled to a repulsiveO(p) scalar field. Regular solutions are obtained for (n=2,p=3,n=3,p=2), and (n=3, p=4). All these solutions have the wormhole topology.     

Gravitational lensing and rotation curve

Based on the geodesic equation in a static spherically symmetric metric we discuss the rotation curve and gravitational lensing. The rotation curve determines one function in the metric without assuming Einstein’s equations. Then lensing is considered in the weak field approximation of general relativity. From the null geodesics we derive the lensing equation. The gravitational potential U(r) which determines the lensing is directly give by the rotation curve U(r) = −v ²(r). This allows to test general relativity on the scale of galaxies where dark matter is relevant.     

Hydrodynamic vacuum sources of dark matter self-generation in an accelerating universe without a Big Bang

We have obtained a generalization of the hydrodynamic theory of vacuum in the context of general relativity. While retaining the Lagrangian character of general relativity, the new theory provides a natural alternative to the view that the singularity is inevitable in general relativity and the theory of a hot Universe. We show that the macroscopic source-sink motion as a whole of ordinary (dark) matter that emerges during the production of particles out of the vacuum can be a new source of gravitational vacuum polarization (determining the variability of the cosmological term in general relativity). We have removed the well-known problems of the cosmological constant by refining the physical nature of dark energy associated precisely with this hydrodynamically initiated variability of the vacuum energy density. A new exact solution of the modified general relativity equations that contains no free (fitting) parameter additional to those available in general relativity has been obtained. It corresponds to the continuous and metric-affecting production of ultralight dark matter particles (with mass m ₀ = (ħ/c ²) $ \sqrt {12\rho _0 k} $ \sqrt {12\rho _0 k} ≈ 3 × 10⁻⁶⁶ g, k is the gravitational constant) out of the vacuum, with its density ρ₀, constant during the exponential expansion of a spatially flat Universe, being retained. This solution is shown to be stable in the regime of cosmological expansion in the time interval −∞ < t < t _max, when t = 0 corresponds to the present epoch and t _max= 2/3H ₀ cΩ_0m ≈ 38 × 10⁹ yr at Ω_0m = ρ₀/ρ_c ≈ 0.28 (H ₀ is the Hubble constant, ρ_c is the critical density). For t > t _max, the solution becomes exponentially unstable and characterizes the inverse process of dark matter particle absorption by the vacuum in the regime of contraction of the Universe. We consider the admissibility of the fact that scalar massive photon pairs can be these dark matter particles. Good quantitative agreement of this exact solution with the cosmological observations of SnIa, SDSS-BAO, and the decrease in the acceleration of the expansion of the Universe has been obtained.     

LETTER: A Simple Quantum Cosmology

A simple and surprisingly realistic model of the origin of the universe can be developed using the Friedmann equation from general relativity, elementary quantum mechanics, and the experimental values of , c, G and the proton mass m _p. The model assumes there are N space dimensions (with N > 6), and the potential constraining the radius r of the invisible N – 3 compact dimensions varies as r ⁴. In this model, the universe has zero total energy and is created from nothing. There is no initial singularity. If space-time is eleven dimensional, as required by M theory, the scalar field corresponding to the size of the compact dimensions inflates the universe by about 26 orders of magnitude (60 e-folds). If H ₀ = 65 km sec^–1 Mpc^–1, the energy density of the scalar field after inflation results in = 0.68, in agreement with recent COBE and Type SNe Ia supernova data.     

Does a varying speed of light solve the cosmological problems?

We propose a new generalisation of general relativity which incorporates a variation in both the speed of light in vacuum (c) and the gravitational constant (G) and which is both covariant and Lorentz invariant. We solve the generalised Einstein equations for Friedmann universes and show that arbitrary time-variations of c and G never lead to a solution to the flatness, horizon or Λ problems for a theory satisfying the strong energy condition. In order to do so, one needs to construct a theory which does not reduce to the standard one for any choice of time, length and energy units. This can be achieved by breaking a number of invariance principles such as covariance and Lorentz invariance.     

On the two-body problem in general relativity

We consider the two-body problem in post-Newtonian approximations of general relativity. We report the recent results concerning the equations of motion, and the associated Lagrangian formulation, of compact binary systems, at the third post-Newtonian order (∼1/c⁶ beyond the Newtonian acceleration). These equations are necessary when constructing the theoretical templates for searching and analyzing the gravitational-wave signals from inspiralling compact binaries in VIRGO and LISA type experiments.     

Higher Dimensional Extension of Charged and Neutral Fluid Spheres with Pressure

General exact higher-dimensional (n+2), n>2 solutions in general theory of relativity of Einstein-Maxwell field equations for spherically symmetric distribution of charged pressure perfect fluid are expressed in terms of pressure extending 4-dimensional solutions presented by Bijalwan (Astrophys. Space Sci. 2011, doi:). Subsequently, metrics (e ^λ and e ^υ ), matter density and electric intensity are expressible in terms of pressure. Consequently, Pressure is found to be an invertible arbitrary function of ω (=c ₁+c ₂ r ²), where c ₁ and c ₂ (≠0) are arbitrary constants, and r is the radius of star, i.e. p=p(ω). We present a general solution for charged pressure fluid in terms for ω. We list and discuss some old and new solutions which fall in this category. Also, these solutions satisfy barotropic equation of state relating the radial pressure to the energy density. But we noticed that none of these solutions in terms of pressure for charged fluids has a well behaved neutral counter part for a spatial component of metric e ^λ i.e. choosing same spatial component for charged and neutral fluid. To illustrate the approach, we discovered a new solution for extended charged analogues of Schwarzschild interior solution in higher dimensions which is found to be well behaved only for n=2. The maximum mass found to be 1.512 M _Θ with linear dimension 14.964 km. Physical quantities pressure, energy density, red-shift, velocity of sound and p/c ² ρ are well behaved and monotonically decreasing towards the surface while adiabatic index and charge density are monotonically increasing. For brevity we don’t discuss the numerical results in detailed.     

The gravitational field in the wave zone I. The radiating terms of the metric tensor

We consider an asymptotically flat space-time generated by a perfect fluid source of compact spatial support. Using the de Donder gauge conditions, the Einstein equations are reduced to a new form of Poisson-type equations. A formal iterative scheme is set up to solve these equations by expanding the components of the metric tensor in powers ofc ^–1. The coefficient of each power ofc ^–1 depends on the asymptotically retarded timeu andx, y, z and satisfies a Poisson-type equation. Assuming asymptotic flatness the solution is carried out in the first orders. The results are explicit expressions of the metric up to orderc ^–4 in terms of the source functions. These expressions hold over all space-time. A further expansion in powers ofr ^–1 gives the first terms of the metric that contribute to gravitational radiation.     

Consequences of the complex character of the internal symmetry in supersymmetric theories

The consequences of the invariance of the superpotential under the complexificationG ^c of the internal symmetry group on the determination of the possible patterns of symmetry and supersymmetry breaking are established in a globally supersymmetric theory. In particular, in the case of global internal symmetry we show that a vacuum associaated to a pointz, whereG _z ^c G _z ^c is always degenerate with a vacuum associated to a pointz, whereG _z ^c =G _z ^c ; all the other degeneracies of the minimum of the potential on an orbit ofG ^c are also determined and shown to be completely removed when the internal symmetry is gauged. The zeroes of theD-term of a supersymmetric gauge theory are characterized as the points of the closed orbits ofG ^c which are at minimum distance from the origin; at these pointsG _z ^c =G _z ^c . It is rigorously proved that the minimum of the potential is zero if the gradient of the superpotential vanishes somewhere. It is also shown that theD-term necessarily vanishes at the minimum of the potential if the direction of spontaneous supersymmetry breaking is invariant byG.Partially supported by the Swiss National Science Foundation and INFN, Sezione di PadovaOn leave of absence from the Department of Physics of the University of Padova, Italy     

Coordinate-dependent 3 + 1 formulation of the general relativity field equations

This paper presents a coordinate-dependent 3+ 1 decomposition of the general relativity field equations in terms of a scalar potentialc ²[(–g ₄₄)^1/2–1], a vector potentialA _icg _4i/(–g₄₄)^1/2, and the three-space metric _ijg _ij–g_4i g _4j/g ₄₄. The equations are exact and the form of the decomposed equations is valid in any coordinate system.     

Variational techniques on classes of Lagrangians

We present canonical procedures for the manipulation of whole classes of Lagrangians that share the same transformation law and functional dependence but are otherwise arbitrary in functional form, and for the derivation therefrom of generalized conserved quantities. The techniques are demonstrated on the class of scalar density LagrangiansL=L _G+L _EM, whereL _G is a function of the metric and its first and second derivatives andL _EM is a function of the metric and a vector potential and its first derivative, which generate the Einstein-Maxwell equations (without cosmological constant). These procedures should be of interest to those studying alternate formulations of general relativity, those deriving new field theories, and others working with general of modified Lagrangians.     

A simplified form for the Hamiltonian and Lagrangian of the spin-independent gravitational two-body system

In general, the gravitational two-body Hamiltonian, to orderc ^–2, containsGP ²,G (P · r)², andG ² terms. We have previously shown [4–6] that a proper choice of coordinate system enables one to eliminate theG (P · r)² term. We now show that, making use of energy conservation, and coordinate transformations, we can eliminate either of the remaining two terms. In particular, we are able to write down a Hamiltonian and a Lagrangian that contain no mixed potential and kinetic terms.Laboratoire associé au Centre National de la Recherche Scientifique.